Angles Between Infinite Dimensional Subspaces
|
|
- Todd Anderson
- 5 years ago
- Views:
Transcription
1 1 Angles Between Infinite Dimensional Subspaces Abram Jujunashvili Department of Mathematics and Center for Computational Mathematics University of Colorado at Denver and Health Sciences Center Ph.D. Thesis Defense November 14, 2005 Supported by the Lawrence Livermore National Laboratory and by the National Science Foundation
2 2 Introduction In this dissertation, we introduce angles between infinite dimensional subspaces in two different ways, based on: spectra of product of orthogonal projectors that may result, e.g., in a set of angles that fills the whole interval [0, π/2]; Courant-Fischer numbers of operators that results in a finite number of angles or in a monotonically nondecreasing or nonincreasing sequence of angles. Courant-Fischer numbers are defined by the Courant-Fischer (min max, max min) principle. We call the second kind of angles the discrete angles. Connection to the finite dimensional case?
3 3 Main new results A new concept of principal invariant subspaces, which generalizes an existing concept of principal vectors; Lidskiĭ Mirsky Wielandt theorem for Courant Fischer numbers of bounded selfadjoint operators; Majorization for sines/cosines (squared) for discrete angles; Majorization for discrete Ritz values. We generalize some known results in the finite dimensional case (e.g., Knyazev and Argentati [2002, 2005]).
4 4 Outline Angles between subspaces; Estimate for proximity of angles; Discrete angles between subspaces; Estimates for proximity and majorization of discrete angles; Estimates for proximity of Ritz values; Analysis of convergence of domain decomposition methods; Application of angles between subspaces in microarray data analysis.
5 5 Why angles are important? Canonical correlations for stochastic processes; Information retrieval for functional data; Perturbations of subspaces related to Rayleigh Ritz approximations; Analysis of convergence of domain decomposition methods; Information about solution quality in eigensolvers; Microarray data analysis.
6 6 Preliminaries Let H be a (real or complex) Hilbert space, F and G be its proper nontrivial subspaces. A subspace is defined as a closed linear manifold. Let P F and P G be the orthogonal projectors onto F and G, respectively. We denote by B(H) the Banach space of bounded linear operators defined on H with a norm T = sup Tu. u H u =1 For T B(H) we define T = (T T) 1/2, using the positive square root. T U denotes the restriction of the operator T to its invariant subspace U. Σ(T) and Σ p (T) denote the spectrum and point spectrum of the operator T, respectively. Σ p (T) is defined as the set of numbers λ in Σ(T), for which λi T is not one-to-one, where I denotes the identity.
7 7 Angles between subspaces. Main new results: Angles from to ˆΘ(F, G) and between Θ(F, G); Connections: ˆΘ(F, G), ˆΘ(F, G ), etc. and Θ(F, G), Θ(F, G ), etc.; Principal invariant subspaces. Recursive relationships and isometries; Changes in the angles where one of the subspaces varies.
8 8 Definitions and properties of angles Definition A set ˆΘ(F, G) = {θ : θ = arccos(σ), σ 0, σ 2 Σ((P F P G ) F )} [0, π 2 ] is called the set of angles from the subspace F to the subspace G. Definition A set Θ(F, G) = ˆΘ(F, G) ˆΘ(G, F) is called the set of angles between the subspaces F and G. Remark In general ˆΘ(F, G) ˆΘ(G, F), i.e. the set-valued function ˆΘ(F, G) is non-symmetric, but Θ(F, G) is a symmetric function. Connection to the finite dimensional case?
9 9 Definitions of angles in the finite dimensional case Denote q = min{dimf, dimg}. Definition The principal angles θ 1,...,θ q [0, π/2] between F and G are defined recursively for k = 1,...,q by cos(θ k ) = max u F max v G (u, v) = (u k, v k ) subject to u = v = 1, u u i = v v i = 0, i = 1,...,k 1. In the infinite dimensional case this definition gives a finite number of angles (q < ) or infinite sequence of angles (q = ), but can not give a whole interval of angles. We use this approach later to define the discrete angles.
10 10 Definitions and properties of angles (cont.) Theorem 1 ˆΘ(F, G) = π 2 ˆΘ(F, G ). Theorem 1 leads to an equivalent definition of the angles using sines. Equivalent definition The set ˆΘ(F, G) = {θ : θ = arcsin(µ), µ 0 µ 2 Σ((P F P G ) F )} [0, π 2 ] is called the set of angles from the subspace F to the subspace G.
11 11 The connections between ˆΘ(F, G), ˆΘ(F, G ), ˆΘ(G, F), etc. Theorem 2 For any pair of subspaces F and G of H: 1. ˆΘ(F, G) = π 2 ˆΘ(F, G ); 2. ˆΘ(G, F) \ { π 2 } = ˆΘ(F, G) \ { π 2 }; 3. ˆΘ(F, G) \ ({0} { π 2 }) = π 2 {ˆΘ(F, G) \ ({0} { π 2 })}; 4. ˆΘ(F, G ) \ ({0} { π 2 }) = ˆΘ(F, G) \ ({0} { π 2 }); 5. ˆΘ(G, F ) \ {0} = π 2 {ˆΘ(F, G) \ { π 2 }}; 6. ˆΘ(G, F) \ { π 2 } = π 2 {ˆΘ(F, G) \ {0}}; 7. ˆΘ(G, F ) \ {0} = ˆΘ(F, G) \ {0}. Where do differences in the zero and right angles come from?
12 12 The connections between Θ(F, G), Θ(F, G ) and Θ(F, G ) Theorem 3 For any subspaces F and G of H the following equalities Θ(F, G) \ ({0} { π 2 }) = {π 2 Θ(F, G )} \ ({0} { π 2 }), Θ(F, G) \ {0} = Θ(F, G ) \ {0} and Θ(F, G ) \ {0} = Θ(F, G) \ {0} hold.
13 13 Spectra of the sum and the difference of projectors Theorem 4 Σ(P F P G ) \ ({ 1} {0} {1}) = ± sin(θ(f, G)) \ ({ 1} {0} {1}). Theorem 5 Σ(P F + P G ) \ ({0} {1}) = {1 ± cos(θ(f, G))} \ ({0} {1}). P G = C A ; P F (1) = C A or (2) = C A (1) Σ(P F + P G ) = {0} {1} {2}; Θ(F, G) = {0}; 1 ± cos(θ(f, G)) = {0} {2}. (2) Σ(P F + P G ) = {0} {1}; Θ(F, G) = {π/2}; 1 ± cos(θ(f, G)) = {1}.
14 14 Angles between subspaces and the gap gap(f, G) = P F P G = max{ P F P G, P G P F } The classical definition, see the textbooks Akhiezer and Glazman [1981], Gohberg and Kreĭn [1969], Kato [1995]. Theorem 6 min{min{cos 2 (ˆΘ(F, G))}, min{cos 2 (ˆΘ(G, F))}} = 1 (gap(f, G)) 2
15 15 Angles between subspaces and the minimum gap For subspaces F and G of H let us define Deutsch [1995] c(f, G) = sup{ (u, v) u F (F G), u 1, and the minimum gap Kato [1995] γ(f, G) = v G (F G), v 1}, inf u F, u/ G d(u, G) d(u, F G). Theorem 7 c(f, G) = sup { cos ( Θ(F, G) \ {0} )} ; γ(f, G) = inf { sin ( Θ(F, G) \ {0} )}.
16 16 Principal vectors Definition ˆΘ p (F, G) = {θ Θ(F, G) : cos 2 (θ) is an eigenvalue of (P F P G ) F }. Θ p (F, G) = ˆΘ p (F, G) ˆΘ p (G, F). Definition Normalized vectors u = u(θ) F and v = v(θ) G form a pair of principal vectors for subspaces F and G corresponding to the angle θ Θ(F, G), if P F v = cos(θ)u and P G u = cos(θ)v. Definition Assume θ Θ p (F, G) \ { π 2 }. Then, the multiplicity of cos2 (θ) as an eigenvalue of the operator (P F P G ) F, is called the multiplicity of θ. Theorem 8 There exists a pair of principal vectors for subspaces F and G corresponding to a given angle θ Θ(F, G) \ { π 2 } if and only if θ Θ p (F, G) \ { π 2 }. Principal vectors are eigenvectors of (P F P G ) F and (P G P F ) G. What s wrong with π/2?
17 17 Principal vectors (cont.) Theorem 9 (In the finite dimensional case, e.g., Wedin [1983], Knyazev and Argentati [2002]) Let u and v form a pair of principal vectors for subspaces F and G, corresponding to the angle θ Θ p (F, G) \ ({0} { π 2 }). Define u = (v cos(θ)u)/ sin(θ), v = (u cos(θ)v)/ sin(θ). Then u, v are the principal vectors for subspaces F and G, corresponding to the angle π 2 θ; u, v are the principal vectors for subspaces F and G, corresponding to the angle π 2 θ; u, v are the principal vectors for subspaces F and G, corresponding to the angle θ.
18 18 Principal subspaces Definition A pair of subspaces U F, V G is called a pair of principal subspaces for subspaces F and G corresponding to the angle θ Θ(F, G), if (P F P V ) F = cos 2 (θ)p U, (P G P U ) G = cos 2 (θ)p V. Theorem 10 Let U F be an eigenspace of an operator (P F P G ) F, corresponding to the eigenvalue cos 2 (θ), where θ Θ p (F, G) \ { π 2 }, P U be an orthogonal projector onto the subspace U, and V = P G U. Then the operator P V = 1 cos 2 (θ) P GP U P G is an orthogonal projector onto the subspace V.
19 19 Principal subspaces (cont.) Theorem 11 U F and V G form a pair of the principal subspaces for subspaces F and G corresponding to the angle θ Θ(F, G) \ { π 2 }, if and only if U and V are eigenspaces of (P F P G ) F and (P G P F ) G, respectively, corresponding to θ Θ p (F, G) \ { π 2 }. Then Θ(U, V) = Θ p(u, V) = {θ}. Theorem 12 If U(θ), U(φ) F, and V(θ), V(φ) G are the principal subspaces for subspaces F and G corresponding to the angles θ, φ Θ p (F, G) \ { π 2 }, then the following relations hold: P U(θ) P U(φ) = P U(θ) U(φ) ; P V(θ) P V(φ) = P V(θ) V(φ) ; P U(θ) and P V(φ) are mutually orthogonal if θ φ. Otherwise, if θ = φ, then we can choose V(θ) such that P U(θ) P V(θ) = P U(θ) P G ; For given U(θ) we can choose V(θ) such that P V(θ) P U(θ) = P V(θ) P F.
20 20 Principal subspaces (cont.) Theorem 13 Let U and V be the principal subspaces for subspaces F and G, corresponding to the angle θ Θ p (F, G) \ ({0} { π 2 }). Define U = R((P F P V ) F ), V = R((P G P U ) G ). Then U and V are closed and U, V are the principal subspaces for subspaces F and G, corresponding to the angle π 2 θ; U, V are the principal subspaces for subspaces F and G, corresponding to the angle π 2 θ; U, V are the principal subspaces for subspaces F and G, corresponding to the angle θ.
21 21 Principal invariant subspaces Definition A pair of subspaces U F, V G is called a pair of principal invariant subspaces for subspaces F and G, if P F V = U and P G U = V. Theorem 14 Let U be an invariant subspace of the operator (P F P G ) F and P U be an orthogonal projector on this subspace. Let the operator (P F P G ) U be invertible and its inverse be bounded. Then the orthogonal projector P V on the invariant subspace V = P G U of the operator (P G P F ) G is given by P V = P G P U ((P F P G ) U ) 1 P U P G. Theorem 15 Let π 2 / Θ(U, V) for a pair of subspaces U and V. U and V form a pair of principal invariant subspaces for subspaces F and G if and only if U is an invariant subspace for (P F P G ) F and V = P G U.
22 22 Principal invariant subspaces (cont.) Principal invariant subspaces can be determined recursively: Theorem 16 Let U F and V G be a pair of principal invariant subspaces for subspaces F and G, and U U, V V be a pair of the principal invariant subspaces for subspaces U and V. Then U, V form a pair of the principal invariant subspaces for subspaces F, G, and Θ(U, V) Θ(U, V) Θ(F, G).
23 23 Principal invariant subspaces and isometries Theorem 17 Let U and V be a pair of principal invariant subspaces of the subspaces F and G. Let W be a (partial) isometry from the polar decomposition P G P F = W P F P G P F. Then V = W U, U = W V, P V = WP U W and P U = W P V W. Remark If the inequality P F P G < 1 holds, then the assertions of Theorem 17 hold for both choices of W = W and W = Ŵ, where W = P G (I P F + P F P G P F ) 1/2 P F (Riesz and Sz.-Nagy [1990]); Ŵ = [P G P F + (I P G )(I P F )][I (P F P G ) 2 ] 1/2 [1970]; Kato [1995]). (Davis and Kahan
24 24 Spectral decompositions and isometries Theorem 18 Let {E 1 } and {E 2 } be the spectral measures of the operators (P F P G ) F and (P G P F ) G, respectively. Further, let Θ Θ(F, G) \ { π 2 } be the Borel set, P U(Θ) = cos(θ) E 1(dλ) and P V(Θ) = cos(θ) E 2(dλ). Then P V(Θ) = P G { cos(θ) } 1 λ E 1(dλ) P G. Theorem 19 Let E 1 (s) and E 2 (s) be the spectral families of the operators P F P G P F and P G P F P G, respectively, and the polar decomposition of the operator P G P F is given by P G P F = W P F P G P F. Then E 2 (s) = WE 1 (s)w.
25 25 Changes in the angles from to where one of the subspaces varies Hausdorff distance dist(s 1, S 2 ) = max{ sup u S 1 d(u, S 2 ), sup v S 2 d(v, S 1 )}. Theorem 20 (Kato [1995], Pták [1978]). Let A, B B(H) be selfadjoint operators. Then Theorem 21 dist(σ(a), Σ(B)) A B. dist(cos 2 (ˆΘ(F, G)), cos 2 (ˆΘ(F, G))) gap(g, G). Angles are stable with respect to a perturbation of a subspace.
26 26 Discrete angles: estimates and majorization Changes in the smallest discrete angles where the subspaces vary; Generalization of the Lidskiĭ Mirsky Wielandt theorem; Majorization for the sines of largest and the cosines of smallest discrete angles; Majorization for the squared sines and cosines of the discrete angles.
27 27 Preliminaries: Courant Fischer numbers The Courant Fischer numbers for a bounded selfadjoint operator A (Dunford and Schwartz [1988]; Reed and Simon [1978]): from the top: λ k (T) = sup H k H inf u Hk (u, Tu), k = 1, 2,.... dimh k =k u =1 The resulting sequence Λ (T) is ordered nonincreasingly; from the bottom: λ k (T) = inf H k H sup u Hk (u, Tu), k = 1, 2,.... dimh k =k u =1 The resulting sequence Λ (T) is ordered nondecreasingly. Connection to the spectrum of T? Connection to matrices?
28 28 Recursive definition of discrete principal angles Definition Denote q = min{dimf, dimg}. The smallest discrete principal angles θ 1,...,θ q [0, π/2] between F and G are defined recursively for k = 1,...,q by cos(θ k ) = sup u F sup (u, v) = (u k, v k ) v G subject to u = v = 1, (u, u i ) = (v, v i ) = 0, i = 1,...,k 1, provided that the pair of vectors u k, v k exists. If for some k = n the pair u k, v k does not exist, we set θ k = θ n for all k > n. Θ d (F, G) = (θ 1,...,θ q ), q is finite or infinite. As a set {Θ d (F, G)} Θ(F, G). Connection to the finite dimensional case?
29 29 Definition of smallest discrete angles using Courant Fischer numbers Equivalent definition The sequence ( ) ) Θ d (θ (F, G) = k : θ k = arccos λ k ((P FP G ) F ), k = 1, 2,...,q where λ k ((P FP G ) F ) are the Courant-Fischer numbers of the operator (P F P G ) F from the top, is called the sequence of smallest discrete principal angles between subspaces F and G.
30 30 Definition of largest discrete angles using Courant Fischer numbers Definition The sequence ( ) ) Θ d (θ (F, G) = k : θ k = arcsin λ k ((P FP G ) F ), k = 1,...,dimF ( ( ) ) θ k : θ k = arcsin λ k ((P GP F ) G ), k = 1, 2,...,dimG where λ k ((P FP G ) F ) and λ k ((P GP F ) G )are the Courant-Fischer numbers from the top of the operators (P F P G ) F and (P G P F ) G, is called the sequence of largest discrete principal angles between subspaces F and G. (1, 1/2,... ) (π/2, π/2, 1, 1/2,... ) = (1, 1/2,... ). Principal vectors, subspaces and invariant subspaces can be defined in the discrete case and have similar properties.
31 31 Changes in the smallest discrete angles where subspaces vary Theorem 22 Let θ k and ˆθ k be the k-th smallest discrete angles between the subspaces F and G, F and G, respectively, k = 1,...,q, q = min{dimf, dimg}. Then for k = 1,...,q, cos(θ k ) cos(ˆθ k ) κ gap(g, G), where κ = max{cos(θ min {(G + G) G, F}); cos(θ min {(G + G) G, F})}. We were not able to obtain similar result for the cosines of largest discrete angles. The proof is based on s-numbers which are defined only from the top. In the finite dimensional case both results are the same and are proven in Knyazev and Argentati [2002].
32 32 Majorization preliminaries Definition (Markus [1964], Marshall and Olkin [1979]) Assume that r = (r 1, r 2,...) and t = (t 1, t 2,...) are (finite or infinite) bounded sequences of real numbers. We say that t weakly majorizes r and write r w t, if sup π k i=1 r πi sup π k t πj, j=1 where the suprema are taken over all k-element permutations. r and t nonincreasing, r w t; Φ(x) convex, Φ( ) = 0. Then P k j=1 Φ(r j) P k j=1 Φ(t j), (k=1,2,...) (Gohberg and Kreĭn [1969]). Theorem 23 p w r and s w t imply (p s) w (r t). In the finite dimensional case this result is known.
33 33 Generalization of the Lidskiĭ Mirsky Wielandt theorem Theorem 24 Let A, B B(H) be selfadjoint operators on H. Then for any choice of indices 1 i 1 < < i k, for any k 1 k λ i j (A + B) j=1 k λ i j (A) + j=1 k λ j (B). j=1 The Lidskiĭ Mirsky Wielandt theorem is one of the major tools. Corollary 1 Under the assumptions of Theorem 24, λ (A + B) λ (A) w λ (B), λ (A) λ (B) w λ (A B), λ (A) λ (B) w λ (A B).
34 34 Majorization for discrete angles Theorem 25 Let F, G, G be subspaces of H, and P G P G < 1 holds. Assume also that max{dimf G, dimf G} = max{dimf G, dimf G} <. Then sinθ d (F, G) sinθ d (F, G) w sinθ d (G, G) holds, where Θ d (F, G), Θ d (F, G) and Θ d (G, G) are the sequences of largest discrete angles between corresponding subspaces.
35 35 Majorization for discrete angles (cont.) Theorem 26 Let F, G, G be subspaces of H, P G P G < 1, and dimf G <, dimf G <, dimf G = dimf G <. Then cos Θ d (F, G) cos Θ d (F, G) w sinθ d (G, G), holds, where Θ d (F, G) and Θ d (F, G) are the sequences of smallest discrete angles and Θ d (G, G) is the sequence of largest discrete angles between corresponding subspaces. We were not able to obtain the result of Theorem 25 for the sines of smallest angles and the result of Theorem 26 for the cosines of largest angles. In finite dimensional case both results are known Knyazev and Argentati [2005].
36 36 Majorization for discrete angles (cont.) Theorem 27 Under the assumptions of Theorem 25, sin 2 Θ d (F, G) sin2 Θ d (F, G) w sinθ d (G, G), cos 2 Θ d (F, G) cos2 Θ d (F, G) w sinθ d (G, G), and under the assumptions of Theorem 26, cos 2 Θ d (F, G) cos2 Θ d (F, G) w sinθ d (G, G), sin 2 Θ d (F, G) sin2 Θ d (F, G) w sinθ d (G, G).
37 37 Estimates for proximity of Ritz values Estimates for proximity of the sets of Ritz values; Majorization for Ritz values in the discrete case. Definition Let X H be a subspace of the Hilbert space H. Then Σ((P X A) X ) is called the set of Ritz values of an operator A with respect to the trial subspace X. Connection to the angles?
38 38 Extending to an orthogonal projector The first step Theorem 28 (Halmos [1969]; Riesz and Sz.-Nagy [1990]) A B(H) selfadjoint nonnegative contraction. Then Â, defined on H H as A A(I A) Â = A(I A) I A is an orthogonal projector.
39 39 Connection of angles to Ritz values The second step Theorem 29 The Ritz values of A and  are the same: Σ((P X A) X ) = Σ((P ˆXÂ) ˆX), ˆX = X 0 Ĥ = H H H. 0 The third step: define subspace Ẑ s.t. P Ẑ =  Σ((P X A) X ) = Σ((P ˆXPẐ) ˆX) = cos 2 (Θ( ˆX, Ẑ)). For the finite dimensional case see Knyazev and Argentati [2005].
40 40 Estimates for proximity of the sets of Ritz values Theorem 30 Let A B(H) be a selfadjoint operator and X and Y be subspaces of H. Then the following inequality dist(σ(p X AP X ), Σ(P Y AP Y )) (M m) P X P Y holds, where m and M are the bounds of the spectrum of A. Theorem 31 Assume A and B are linear bounded operators on a Hilbert space H and the equalities Σ(AA ) = Σ(A A) and Σ(BB ) = Σ(B B) hold. Then the following inequality holds: dist(σ( A ), Σ( B )) A B.
41 41 Majorization for Ritz values in discrete case Definition Let A B(H) be selfadjoint and X be a subspace of H. The sequence of Courant-Fischer numbers Λ ((P X A) X ) is called the sequence of the discrete Ritz values from the bottom of the operator A with respect to the trial subspace X. Theorem 32 Let A B(H) be selfadjoint and X and Y be subspaces of H and codimx = codimy <. Then Λ ((P X A) X ) Λ ((P Y A) Y ) w (M m) sinθ d (X, Y), where m and M are the greatest lower and least upper bounds of Σ(A). For Ritz values from the top we use an identity Λ ((P X A) X ) = Λ ( (P X A) X ).
42 Analysis of convergence of domain decomposition methods u v dx = 1 0 fvdx, v H = H 1 0([0, 1]); u H. Ω 1 = [0, α]; Ω 2 = [β,1]; 0 < β < α < 1; an overlap [β, α]. F = {f H f(x) = 0, x [α,1]} G = {g H g(x) = 0, x [0, β]} e k+1 = P G P F e k, k = 0, 1,...; (I P F P G P F )e = 0. e k+1 = [I (P F + P G )]e k, k = 0, 1,...; (P F + P G )e = 0.
43 43 Let θ = arccos Theorem 33 Analysis of the convergence (cont.) ( β(1 α) α(1 β) ). Σ(I P F P G P F ) = Σ(I P G P F ) = {sin 2 (θ)} {1}. Following Bjørstad and Mandel [1991], Vidav [1977], we prove Theorem 34 Σ(P F + P G ) = {1, 1 ± cos(θ), 2}. The conjugate gradient method converges for multiplicative Schwarz in two iterations and for additive Schwarz in four iterations.
44 44 Application of angles between subspaces in microarray data analysis The angles between subspaces are closely related to canonical correlations in statistics. We propose a new application of the angles between subspaces for microarray data analysis. For this purpose, we develop a public supplement to MALTAB s BioInformatics toolbox that performs single-array and comparison analysis for Affymetrix microarray data analogously to the Affymetrix software. We propose and implement a novel multi-comparison analysis of the Affymetrix microarray data using angles between subspaces.
45 45 References N. I. Akhiezer and I. M. Glazman. Theory of linear operators in Hilbert space. Vol. I, volume 9 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, Mass., ISBN X. Translated from the third Russian edition by E. R. Dawson, Translation edited by W. N. Everitt. Petter E. Bjørstad and Jan Mandel. On the spectra of sums of orthogonal projections with applications to parallel computing. BIT, 31(1):76 88, ISSN Chandler Davis and W. M. Kahan. The rotation of eigenvectors by a perturbation. III. SIAM J. Numer. Anal., 7:1 46, ISSN Frank Deutsch. The angle between subspaces of a Hilbert space. In
46 46 Approximation theory, wavelets and applications (Maratea, 1994), volume 454 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages Kluwer Acad. Publ., Dordrecht, Nelson Dunford and Jacob T. Schwartz. Linear operators. Part II. Wiley Classics Library. John Wiley & Sons Inc., New York, ISBN Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication. I. C. Gohberg and M. G. Kreĭn. Introduction to the theory of linear nonselfadjoint operators. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18. American Mathematical Society, Providence, R.I., 1969.
47 47 P. R. Halmos. Two subspaces. Trans. Amer. Math. Soc., 144: , ISSN Tosio Kato. Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, ISBN X. Reprint of the 1980 edition. A. V. Knyazev and M. E. Argentati. Principal angles between subspaces in an A-based scalar product: algorithms and perturbation estimates. SIAM J. Sci. Comput., 23(6): (electronic), ISSN A. V. Knyazev and M. E. Argentati. Majorization for changes in angles between subspaces, Ritz values, and graph laplacian spectra. SIAM Review, accepted, A. S. Markus. Eigenvalues and singular values of the sum and product of
48 48 linear operators. Uspehi Mat. Nauk, 19(4 (118)):93 123, ISSN Albert W. Marshall and Ingram Olkin. Inequalities: theory of majorization and its applications, volume 143 of Mathematics in Science and Engineering. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, ISBN Vlastimil Pták. Perturbations and continuity of the spectrum. In Proceedings of the International Conference on Operator Algebras, Ideals, and their Applications in Theoretical Physics (Leipzig, 1977), pages , Leipzig, Teubner. Michael Reed and Barry Simon. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York, ISBN
49 49 Frigyes Riesz and Béla Sz.-Nagy. Functional analysis. Dover Books on Advanced Mathematics. Dover Publications Inc., New York, ISBN Translated from the second French edition by Leo F. Boron, Reprint of the 1955 original. Ivan Vidav. The norm of the sum of two projections. Proc. Amer. Math. Soc., 65(2): , ISSN P. A. Wedin. On angles between subspaces of a finite-dimensional inner product space. In Bo Kȧgström and Axel Ruhe, editors, Matrix Pencils. Proceedings of the Conference Held at Pite Havsbad, March 22 24, 1982, pages Springer Verlag, Berlin, 1983.
Angles Between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods
Angles Between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods Andrew Knyazev a,,1, Abram Jujunashvili a, Merico Argentati a a Department of Mathematical
More informationANGLES BETWEEN INFINITE-DIMENSIONAL SUBSPACES. Abram Jujunashvili. M.S., Tbilisi State University, Tbilisi, Georgia, 1978
ANGLES BETWEEN INFINITE-DIMENSIONAL SUBSPACES by Abram Jujunashvili M.S., Tbilisi State University, Tbilisi, Georgia, 1978 Ph.D., Optimization, Georgian Academy of Sciences, Tbilisi, 1984 A thesis submitted
More informationANGLES BETWEEN INFINITE DIMENSIONAL SUBSPACES WITH APPLICATIONS TO THE RAYLEIGH-RITZ AND ALTERNATING PROJECTORS METHODS
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0894-0347(XX)0000-0 ANGLES BETWEEN INFINITE DIMENSIONAL SUBSPACES WITH APPLICATIONS TO THE RAYLEIGH-RITZ AND ALTERNATING
More informationMajorization for Changes in Ritz Values and Canonical Angles Between Subspaces (Part I and Part II)
1 Majorization for Changes in Ritz Values and Canonical Angles Between Subspaces (Part I and Part II) Merico Argentati (speaker), Andrew Knyazev, Ilya Lashuk and Abram Jujunashvili Department of Mathematics
More informationMAJORIZATION FOR CHANGES IN ANGLES BETWEEN SUBSPACES, RITZ VALUES, AND GRAPH LAPLACIAN SPECTRA
SIAM J. MATRIX ANAL. APPL. Vol.?, No.?, pp.?? c 2006 Society for Industrial and Applied Mathematics MAJORIZATION FOR CHANGES IN ANGLES BETWEEN SUBSPACES, RITZ VALUES, AND GRAPH LAPLACIAN SPECTRA ANDREW
More informationMajorization for Changes in Angles Between Subspaces, Ritz values, and graph Laplacian spectra
University of Colorado at Denver and Health Sciences Center Majorization for Changes in Angles Between Subspaces, Ritz values, and graph Laplacian spectra A. V. Knyazev and M. E. Argentati September 2005
More informationRayleigh-Ritz majorization error bounds with applications to FEM and subspace iterations
1 Rayleigh-Ritz majorization error bounds with applications to FEM and subspace iterations Merico Argentati and Andrew Knyazev (speaker) Department of Applied Mathematical and Statistical Sciences Center
More informationPrincipal Angles Between Subspaces and Their Tangents
MITSUBISI ELECTRIC RESEARC LABORATORIES http://wwwmerlcom Principal Angles Between Subspaces and Their Tangents Knyazev, AV; Zhu, P TR2012-058 September 2012 Abstract Principal angles between subspaces
More informationAPPROXIMATION OF MOORE-PENROSE INVERSE OF A CLOSED OPERATOR BY A SEQUENCE OF FINITE RANK OUTER INVERSES
APPROXIMATION OF MOORE-PENROSE INVERSE OF A CLOSED OPERATOR BY A SEQUENCE OF FINITE RANK OUTER INVERSES S. H. KULKARNI AND G. RAMESH Abstract. Let T be a densely defined closed linear operator between
More informationMAJORIZATION FOR CHANGES IN ANGLES BETWEEN SUBSPACES, RITZ VALUES, AND GRAPH LAPLACIAN SPECTRA
SIAM J. MATRIX ANAL. APPL. Vol. 29, No. 1, pp. 15 32 c 2006 Society for Industrial and Applied Mathematics MAJORIZATION FOR CHANGES IN ANGLES BETWEEN SUBSPACES, RITZ VALUES, AND GRAPH LAPLACIAN SPECTRA
More informationNEW ESTIMATES FOR RITZ VECTORS
MATHEMATICS OF COMPUTATION Volume 66, Number 219, July 1997, Pages 985 995 S 0025-5718(97)00855-7 NEW ESTIMATES FOR RITZ VECTORS ANDREW V. KNYAZEV Abstract. The following estimate for the Rayleigh Ritz
More informationPerturbation Theory for Self-Adjoint Operators in Krein spaces
Perturbation Theory for Self-Adjoint Operators in Krein spaces Carsten Trunk Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany E-mail: carsten.trunk@tu-ilmenau.de
More informationAn Iterative Procedure for Solving the Riccati Equation A 2 R RA 1 = A 3 + RA 4 R. M.THAMBAN NAIR (I.I.T. Madras)
An Iterative Procedure for Solving the Riccati Equation A 2 R RA 1 = A 3 + RA 4 R M.THAMBAN NAIR (I.I.T. Madras) Abstract Let X 1 and X 2 be complex Banach spaces, and let A 1 BL(X 1 ), A 2 BL(X 2 ), A
More information1 Quasi-definite matrix
1 Quasi-definite matrix The matrix H is a quasi-definite matrix, if there exists a permutation matrix P such that H qd P T H11 H HP = 1 H1, 1) H where H 11 and H + H1H 11 H 1 are positive definite. This
More informationBOUNDARY VALUE PROBLEMS IN KREĬN SPACES. Branko Ćurgus Western Washington University, USA
GLASNIK MATEMATIČKI Vol. 35(55(2000, 45 58 BOUNDARY VALUE PROBLEMS IN KREĬN SPACES Branko Ćurgus Western Washington University, USA Dedicated to the memory of Branko Najman. Abstract. Three abstract boundary
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 1, pp. 1-11, 8. Copyright 8,. ISSN 168-961. MAJORIZATION BOUNDS FOR RITZ VALUES OF HERMITIAN MATRICES CHRISTOPHER C. PAIGE AND IVO PANAYOTOV Abstract.
More informationOn compact operators
On compact operators Alen Aleanderian * Abstract In this basic note, we consider some basic properties of compact operators. We also consider the spectrum of compact operators on Hilbert spaces. A basic
More informationDAVIS WIELANDT SHELLS OF NORMAL OPERATORS
DAVIS WIELANDT SHELLS OF NORMAL OPERATORS CHI-KWONG LI AND YIU-TUNG POON Dedicated to Professor Hans Schneider for his 80th birthday. Abstract. For a finite-dimensional operator A with spectrum σ(a), the
More informationSingular Value Inequalities for Real and Imaginary Parts of Matrices
Filomat 3:1 16, 63 69 DOI 1.98/FIL16163C Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Singular Value Inequalities for Real Imaginary
More informationON ANGLES BETWEEN SUBSPACES OF INNER PRODUCT SPACES
ON ANGLES BETWEEN SUBSPACES OF INNER PRODUCT SPACES HENDRA GUNAWAN AND OKI NESWAN Abstract. We discuss the notion of angles between two subspaces of an inner product space, as introduced by Risteski and
More informationMathematik-Bericht 2010/5. An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schrödinger operators
Mathematik-Bericht 2010/5 An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schrödinger operators M. Hansmann Juni 2010 Institut für Mathematik, TU Clausthal, Erzstraße 1, D-38678
More informationCESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE
CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS Abstract. In this article we study the Cesàro operator C(f)() = d, and its companion operator
More informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J. Math. Anal. 6 (2012), no. 1, 139 146 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) www.emis.de/journals/bjma/ AN EXTENSION OF KY FAN S DOMINANCE THEOREM RAHIM ALIZADEH
More informationQUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE. Denitizable operators in Krein spaces have spectral properties similar to those
QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE BRANKO CURGUS and BRANKO NAJMAN Denitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces.
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More informationSchur functions. J. Rovnyak and H. S. V. de Snoo
Schur functions J. Rovnyak and H. S. V. de Snoo The Schur class in complex analysis is the set of holomorphic functions S(z) which are defined and satisfy S(z) 1 on the unit disk D = {z : z < 1} in the
More informationSufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems
Electronic Journal of Differential Equations, Vol. 200(200), No. 74, pp. 0. ISSN: 072-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Sufficient conditions
More informationOPERATOR SEMIGROUPS. Lecture 3. Stéphane ATTAL
Lecture 3 OPRATOR SMIGROUPS Stéphane ATTAL Abstract This lecture is an introduction to the theory of Operator Semigroups and its main ingredients: different types of continuity, associated generator, dual
More informationMAIN ARTICLES. In present paper we consider the Neumann problem for the operator equation
Volume 14, 2010 1 MAIN ARTICLES THE NEUMANN PROBLEM FOR A DEGENERATE DIFFERENTIAL OPERATOR EQUATION Liparit Tepoyan Yerevan State University, Faculty of mathematics and mechanics Abstract. We consider
More informationISOLATED SEMIDEFINITE SOLUTIONS OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATION
ISOLATED SEMIDEFINITE SOLUTIONS OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATION Harald K. Wimmer 1 The set of all negative-semidefinite solutions of the CARE A X + XA + XBB X C C = 0 is homeomorphic
More informationON THE UNILATERAL SHIFT AND COMMUTING CONTRACTIONS
ON THE UNILATERAL SHIFT AND COMMUTING CONTRACTIONS JUSTUS K. MILE Kiriri Women s University of Science & Technology, P. O. Box 49274-00100 Nairobi, Kenya Abstract In this paper, we discuss the necessary
More informationMeans of unitaries, conjugations, and the Friedrichs operator
J. Math. Anal. Appl. 335 (2007) 941 947 www.elsevier.com/locate/jmaa Means of unitaries, conjugations, and the Friedrichs operator Stephan Ramon Garcia Department of Mathematics, Pomona College, Claremont,
More informationA DECOMPOSITION THEOREM FOR FRAMES AND THE FEICHTINGER CONJECTURE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 A DECOMPOSITION THEOREM FOR FRAMES AND THE FEICHTINGER CONJECTURE PETER G. CASAZZA, GITTA KUTYNIOK,
More informationClasses of Linear Operators Vol. I
Classes of Linear Operators Vol. I Israel Gohberg Seymour Goldberg Marinus A. Kaashoek Birkhäuser Verlag Basel Boston Berlin TABLE OF CONTENTS VOLUME I Preface Table of Contents of Volume I Table of Contents
More informationComplex symmetric operators
Complex symmetric operators Stephan Ramon Garcia 1 Complex symmetric operators This section is a brief introduction to complex symmetric operators, a certain class of Hilbert space operators which arise
More informationSELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More informationInequalities in Hilbert Spaces
Inequalities in Hilbert Spaces Jan Wigestrand Master of Science in Mathematics Submission date: March 8 Supervisor: Eugenia Malinnikova, MATH Norwegian University of Science and Technology Department of
More informationarxiv: v1 [math.fa] 12 Oct 2016
UNITARILY INVARIANT NORM INEQUALITIES FOR ELEMENTARY OPERATORS INVOLVING G 1 OPERATORS FUAD KITTANEH, MOHAMMAD SAL MOSLEHIAN, AND MOHAMMAD SABABHEH arxiv:161.3869v1 [math.fa] 1 Oct 16 Abstract. In this
More informationCOMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE
COMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE MICHAEL LAUZON AND SERGEI TREIL Abstract. In this paper we find a necessary and sufficient condition for two closed subspaces, X and Y, of a Hilbert
More informationCOMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED TIME SCALES. Hüseyin Tuna
Indian J. Pure Appl. Math., 47(3): 535-544, September 2016 c Indian National Science Academy DOI: 10.1007/s13226-016-0196-1 COMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More informationFunctional Analysis. James Emery. Edit: 8/7/15
Functional Analysis James Emery Edit: 8/7/15 Contents 0.1 Green s functions in Ordinary Differential Equations...... 2 0.2 Integral Equations........................ 2 0.2.1 Fredholm Equations...................
More informationOn the simplest expression of the perturbed Moore Penrose metric generalized inverse
Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated
More informationINVESTIGATING THE NUMERICAL RANGE AND Q-NUMERICAL RANGE OF NON SQUARE MATRICES. Aikaterini Aretaki, John Maroulas
Opuscula Mathematica Vol. 31 No. 3 2011 http://dx.doi.org/10.7494/opmath.2011.31.3.303 INVESTIGATING THE NUMERICAL RANGE AND Q-NUMERICAL RANGE OF NON SQUARE MATRICES Aikaterini Aretaki, John Maroulas Abstract.
More informationOperators with numerical range in a closed halfplane
Operators with numerical range in a closed halfplane Wai-Shun Cheung 1 Department of Mathematics, University of Hong Kong, Hong Kong, P. R. China. wshun@graduate.hku.hk Chi-Kwong Li 2 Department of Mathematics,
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationOn Kusuoka Representation of Law Invariant Risk Measures
MATHEMATICS OF OPERATIONS RESEARCH Vol. 38, No. 1, February 213, pp. 142 152 ISSN 364-765X (print) ISSN 1526-5471 (online) http://dx.doi.org/1.1287/moor.112.563 213 INFORMS On Kusuoka Representation of
More informationUpper triangular forms for some classes of infinite dimensional operators
Upper triangular forms for some classes of infinite dimensional operators Ken Dykema, 1 Fedor Sukochev, 2 Dmitriy Zanin 2 1 Department of Mathematics Texas A&M University College Station, TX, USA. 2 School
More information^-,({«}))=ic^if<iic/n^ir=iic/, where e(n) is the sequence defined by e(n\m) = 8^, (the Kronecker delta).
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 70, Number 1, June 1978 COMPOSITION OPERATOR ON F AND ITS ADJOINT R. K. SINGH AND B. S. KOMAL Abstract. A necessary and sufficient condition for
More informationPerturbations of functions of diagonalizable matrices
Electronic Journal of Linear Algebra Volume 20 Volume 20 (2010) Article 22 2010 Perturbations of functions of diagonalizable matrices Michael I. Gil gilmi@bezeqint.net Follow this and additional works
More informationThe Hopf argument. Yves Coudene. IRMAR, Université Rennes 1, campus beaulieu, bat Rennes cedex, France
The Hopf argument Yves Coudene IRMAR, Université Rennes, campus beaulieu, bat.23 35042 Rennes cedex, France yves.coudene@univ-rennes.fr slightly updated from the published version in Journal of Modern
More informationThe Norm and Modulus of a Foguel Operator
The Norm and Modulus of a Foguel Operator STEPHAN RAMON GARCIA ABSTRACT. We develop a method for calculating the norm and the spectrum of the modulus of a Foguel operator. In many cases, the norm can be
More informationA DECOMPOSITION OF E 0 -SEMIGROUPS
A DECOMPOSITION OF E 0 -SEMIGROUPS Remus Floricel Abstract. Any E 0 -semigroup of a von Neumann algebra can be uniquely decomposed as the direct sum of an inner E 0 -semigroup and a properly outer E 0
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationConvergence rate estimates for the gradient differential inclusion
Convergence rate estimates for the gradient differential inclusion Osman Güler November 23 Abstract Let f : H R { } be a proper, lower semi continuous, convex function in a Hilbert space H. The gradient
More informationON OPERATORS WITH AN ABSOLUTE VALUE CONDITION. In Ho Jeon and B. P. Duggal. 1. Introduction
J. Korean Math. Soc. 41 (2004), No. 4, pp. 617 627 ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION In Ho Jeon and B. P. Duggal Abstract. Let A denote the class of bounded linear Hilbert space operators with
More informationGeneralized Absolute Values and Polar Decompositions of a Bounded Operator
Integr. Equ. Oper. Theory 71 (2011), 151 160 DOI 10.1007/s00020-011-1896-x Published online July 30, 2011 c The Author(s) This article is published with open access at Springerlink.com 2011 Integral Equations
More informationOn m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry
On m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry Ognjen Milatovic Department of Mathematics and Statistics University of North Florida Jacksonville, FL 32224 USA. Abstract
More informationUsing the Karush-Kuhn-Tucker Conditions to Analyze the Convergence Rate of Preconditioned Eigenvalue Solvers
Using the Karush-Kuhn-Tucker Conditions to Analyze the Convergence Rate of Preconditioned Eigenvalue Solvers Merico Argentati University of Colorado Denver Joint work with Andrew V. Knyazev, Klaus Neymeyr
More informationTitle: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on
Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on manifolds. Author: Ognjen Milatovic Department Address: Department
More informationIs there life after the Lanczos method? What is LOBPCG?
1 Is there life after the Lanczos method? What is LOBPCG? Andrew V. Knyazev Department of Mathematics and Center for Computational Mathematics University of Colorado at Denver SIAM ALA Meeting, July 17,
More informationMath 408 Advanced Linear Algebra
Math 408 Advanced Linear Algebra Chi-Kwong Li Chapter 4 Hermitian and symmetric matrices Basic properties Theorem Let A M n. The following are equivalent. Remark (a) A is Hermitian, i.e., A = A. (b) x
More informationRemarks on the Rademacher-Menshov Theorem
Remarks on the Rademacher-Menshov Theorem Christopher Meaney Abstract We describe Salem s proof of the Rademacher-Menshov Theorem, which shows that one constant works for all orthogonal expansions in all
More informationA NOTE ON QUASI ISOMETRIES II. S.M. Patel Sardar Patel University, India
GLASNIK MATEMATIČKI Vol. 38(58)(2003), 111 120 A NOTE ON QUASI ISOMETRIES II S.M. Patel Sardar Patel University, India Abstract. An operator A on a complex Hilbert space H is called a quasi-isometry if
More informationSubstrictly Cyclic Operators
Substrictly Cyclic Operators Ben Mathes dbmathes@colby.edu April 29, 2008 Dedicated to Don Hadwin Abstract We initiate the study of substrictly cyclic operators and algebras. As an application of this
More informationON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS. Srdjan Petrović
ON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS Srdjan Petrović Abstract. In this paper we show that every power bounded operator weighted shift with commuting normal weights is similar to a contraction.
More informationA f = A f (x)dx, 55 M F ds = M F,T ds, 204 M F N dv n 1, 199 !, 197. M M F,N ds = M F ds, 199 (Δ,')! = '(Δ)!, 187
References 1. T.M. Apostol; Mathematical Analysis, 2nd edition, Addison-Wesley Publishing Co., Reading, Mass. London Don Mills, Ont., 1974. 2. T.M. Apostol; Calculus Vol. 2: Multi-variable Calculus and
More informationNEW A PRIORI FEM ERROR ESTIMATES FOR EIGENVALUES
NEW A PRIORI FEM ERROR ESTIMATES FOR EIGENVALUES ANDREW V. KNYAZEV AND JOHN E. OSBORN Abstract. We analyze the Ritz method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates.
More informationOn Riesz-Fischer sequences and lower frame bounds
On Riesz-Fischer sequences and lower frame bounds P. Casazza, O. Christensen, S. Li, A. Lindner Abstract We investigate the consequences of the lower frame condition and the lower Riesz basis condition
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationDragan S. Djordjević. 1. Motivation
ON DAVIS-AAN-WEINERGER EXTENSION TEOREM Dragan S. Djordjević Abstract. If R = where = we find a pseudo-inverse form of all solutions W = W such that A = R where A = W and R. In this paper we extend well-known
More informationThe Knaster problem and the geometry of high-dimensional cubes
The Knaster problem and the geometry of high-dimensional cubes B. S. Kashin (Moscow) S. J. Szarek (Paris & Cleveland) Abstract We study questions of the following type: Given positive semi-definite matrix
More informationANGLES BETWEEN SUBSPACES AND THE RAYLEIGH-RITZ METHOD. Peizhen Zhu. M.S., University of Colorado Denver, A thesis submitted to the
ANGLES BETWEEN SUBSPACES AND THE RAYLEIGH-RITZ METHOD by Peizhen Zhu M.S., University of Colorado Denver, 2009 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in
More informationInvariant subspaces for operators whose spectra are Carathéodory regions
Invariant subspaces for operators whose spectra are Carathéodory regions Jaewoong Kim and Woo Young Lee Abstract. In this paper it is shown that if an operator T satisfies p(t ) p σ(t ) for every polynomial
More informationc 2009 Society for Industrial and Applied Mathematics
SIAM J. MATRIX ANAL. APPL. Vol.??, No.?, pp.?????? c 2009 Society for Industrial and Applied Mathematics GRADIENT FLOW APPROACH TO GEOMETRIC CONVERGENCE ANALYSIS OF PRECONDITIONED EIGENSOLVERS ANDREW V.
More informationSpectra of Adjacency and Laplacian Matrices
Spectra of Adjacency and Laplacian Matrices Definition: University of Alicante (Spain) Matrix Computing (subject 3168 Degree in Maths) 30 hours (theory)) + 15 hours (practical assignment) Contents 1. Spectra
More informationSTRUCTURAL AND SPECTRAL PROPERTIES OF k-quasi- -PARANORMAL OPERATORS. Fei Zuo and Hongliang Zuo
Korean J Math (015), No, pp 49 57 http://dxdoiorg/1011568/kjm01549 STRUCTURAL AND SPECTRAL PROPERTIES OF k-quasi- -PARANORMAL OPERATORS Fei Zuo and Hongliang Zuo Abstract For a positive integer k, an operator
More informationA Dykstra-like algorithm for two monotone operators
A Dykstra-like algorithm for two monotone operators Heinz H. Bauschke and Patrick L. Combettes Abstract Dykstra s algorithm employs the projectors onto two closed convex sets in a Hilbert space to construct
More informationSOME PROPERTIES OF ESSENTIAL SPECTRA OF A POSITIVE OPERATOR
SOME PROPERTIES OF ESSENTIAL SPECTRA OF A POSITIVE OPERATOR E. A. ALEKHNO (Belarus, Minsk) Abstract. Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σ ew (T ) of the
More informatione jk :=< e k, e j >, e[t ] e = E 1 ( e [T ] e ) h E.
onjb.tex 14.02.23 Orthonormal Jordan bases in finite dimensional Hilbert spaces 1. Introduction and terminology The aim of this paper is twofold: first, we give a necessary and sufficient condition for
More informationdominant positive-normal. In view of (1), it is very natural to study the properties of positivenormal
Bull. Korean Math. Soc. 39 (2002), No. 1, pp. 33 41 ON POSITIVE-NORMAL OPERATORS In Ho Jeon, Se Hee Kim, Eungil Ko, and Ji Eun Park Abstract. In this paper we study the properties of positive-normal operators
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationThe Drazin inverses of products and differences of orthogonal projections
J Math Anal Appl 335 7 64 71 wwwelseviercom/locate/jmaa The Drazin inverses of products and differences of orthogonal projections Chun Yuan Deng School of Mathematics Science, South China Normal University,
More informationSOLVING MESH EIGENPROBLEMS WITH MULTIGRID EFFICIENCY
SOLVING MESH EIGENPROBLEMS WITH MULTIGRID EFFICIENCY KLAUS NEYMEYR ABSTRACT. Multigrid techniques can successfully be applied to mesh eigenvalue problems for elliptic differential operators. They allow
More informationTHREE SPACES PROBLEM FOR LYAPUNOV THEOREM ON VECTOR MEASURE. V. M. Kadets, O. I. Vladimirskaya
Serdica Math. J. 24 (998), 45-52 THREE SPACES PROBLEM FOR LYAPUNOV THEOREM ON VECTOR MEASURE V. M. Kadets, O. I. Vladimirskaya Communicated by G. Godefroy Abstract. It is proved that a Banach space X has
More informationBEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction
Bull Korean Math Soc 43 (2006), No 2, pp 377 387 BEST APPROXIMATIONS AND ORTHOGONALITIES IN -INNER PRODUCT SPACES Seong Sik Kim* and Mircea Crâşmăreanu Abstract In this paper, some characterizations of
More informationTHE SPECTRAL DIAMETER IN BANACH ALGEBRAS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume»!. Number 1, Mav 1984 THE SPECTRAL DIAMETER IN BANACH ALGEBRAS SANDY GRABINER1 Abstract. The element a is in the center of the Banach algebra A modulo
More informationBASIC MATRIX PERTURBATION THEORY
BASIC MATRIX PERTURBATION THEORY BENJAMIN TEXIER Abstract. In this expository note, we give the proofs of several results in finitedimensional matrix perturbation theory: continuity of the spectrum, regularity
More informationClassical stuff - title to be changed later
CHAPTER 1 Classical stuff - title to be changed later 1. Positive Definite Kernels To start with something simple and elegant, we choose positive definite kernels which appear at every corner in functional
More informationWEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE
Fixed Point Theory, Volume 6, No. 1, 2005, 59-69 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.htm WEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE YASUNORI KIMURA Department
More informationPatryk Pagacz. Characterization of strong stability of power-bounded operators. Uniwersytet Jagielloński
Patryk Pagacz Uniwersytet Jagielloński Characterization of strong stability of power-bounded operators Praca semestralna nr 3 (semestr zimowy 2011/12) Opiekun pracy: Jaroslav Zemanek CHARACTERIZATION OF
More informationNonlinear stabilization via a linear observability
via a linear observability Kaïs Ammari Department of Mathematics University of Monastir Joint work with Fathia Alabau-Boussouira Collocated feedback stabilization Outline 1 Introduction and main result
More informationA Brief Introduction to Functional Analysis
A Brief Introduction to Functional Analysis Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 5, 2007 Definition 1. An algebra A is a vector space over C with
More informationTHE BOUNDEDNESS BELOW OF 2 2 UPPER TRIANGULAR OPERATOR MATRICES. In Sung Hwang and Woo Young Lee 1
THE BOUNDEDNESS BELOW OF 2 2 UPPER TRIANGULAR OPERATOR MATRICES In Sung Hwang and Woo Young Lee 1 When A LH and B LK are given we denote by M C an operator acting on the Hilbert space H K of the form M
More informationChapter 3 Transformations
Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases
More informationTHE ANGLE OF AN OPERATOR AND POSITIVE OPERATOR PRODUCTS 1
THE ANGLE OF AN OPERATOR AND POSITIVE OPERATOR PRODUCTS 1 BY KARL GUSTAFSON Communicated by F. E. Browder, December 5, 1967 1. Introduction. Let -4^0 and JB^O be two positive bounded selfadjoint operators.
More informationRITZ VALUE BOUNDS THAT EXPLOIT QUASI-SPARSITY
RITZ VALUE BOUNDS THAT EXPLOIT QUASI-SPARSITY ILSE C.F. IPSEN Abstract. Absolute and relative perturbation bounds for Ritz values of complex square matrices are presented. The bounds exploit quasi-sparsity
More informationREMARKS ON THE MEMBRANE AND BUCKLING EIGENVALUES FOR PLANAR DOMAINS. Leonid Friedlander
REMARKS ON THE MEMBRANE AND BUCKLING EIGENVALUES FOR PLANAR DOMAINS Leonid Friedlander Abstract. I present a counter-example to the conjecture that the first eigenvalue of the clamped buckling problem
More information