Linear maps preserving AN -operators

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Linear maps preserving AN -operators (Ritsumeikan University) co-author: Golla Ramesh (Indian Instiute of Technology Hyderabad) Numerical Range and Numerical Radii Max-Planck-Institute MPQ June 14, 2018

Outline 1 Linear preserving problems 2 N -operators 3 Properties for N -operators 4 Examples related to N -operators 5 AN -operators 6 Necessary conditions for positive AN -operators 7 Properties for AN -operators 8 Maps preserving AN -operators

Linear preserving problems Let H be a separable, inifinite dimensional Hilbert space and B(H) the set of all bounded operators on H. The following are three typical linear preserving problwms which are pointed out by Chi-Kwong Li-S. Pierce (2001) and Lajos Molnár (2006): Let M be a subspace of B(H).

Linear preserving problems Let H be a separable, inifinite dimensional Hilbert space and B(H) the set of all bounded operators on H. The following are three typical linear preserving problwms which are pointed out by Chi-Kwong Li-S. Pierce (2001) and Lajos Molnár (2006): Let M be a subspace of B(H). 1 Let F be a (scalar-valued, vector-valued, or set valued) function on M, study the linear preservers of F, i.e., thoses linear operators on M such satisfying F (ϕ(a)) = F (A) for all A M. 2 Let S be a subset of M. Characterize thoses linear preservers ϕ on M which satisfy ϕ(s) S, or satisfy A S ϕ(a) S. 3 Let be a relation on M. Characterize thoses linear preservers ϕ on M which satisfy A B ϕ(a) ϕ(b), or satisfy A B ϕ(a) ϕ(b).

In this talk we consider problem 2 in the case that S is the set of all AN -operators.

N -operators Definition Let H and K be separable infinite dimensional Hilbert spaces. Let T B(H, K). T is said to be an N -operator or to satisfy the norm attaining property if there is an element x in the unit sphere of H such that T = Tx. We describe N (H, K) as the set of all N -operators.

N -operators Definition Let H and K be separable infinite dimensional Hilbert spaces. Let T B(H, K). T is said to be an N -operator or to satisfy the norm attaining property if there is an element x in the unit sphere of H such that T = Tx. We describe N (H, K) as the set of all N -operators. When T B(H) and T = T, T is an N -operator if and only if T or T is an eigenvalue of T. Indeed, if T is an N -operator, there exists a unit vector x 0 H such that Tx 0 = T. We may assume that T = 1. Since (I T 2 )x 0, x 0 = 0, (I T 2 )x 0 = 0. Hence, (I + T )(x 0 Tx 0 ) = 0 or (I T )(x 0 + Tx 0 ) = 0. We know, then, that T or T is an eigenvalue of T.

Properties for N -operators The following are proved by Carvajal and Neves (2012): Theorem Let H and K be separable infinite dimensional Hilbert spaces and T B(H, K). The following statements are equivalent. 1 T N (H, K), 2 T N (K, H), 3 T N (H, K), 4 T N (K, H), 5 T 2 N (H, K), 6 T 2 N (K, H),

Properties for N -operators The following are proved by Carvajal and Neves (2012): Theorem Let H and K be separable infinite dimensional Hilbert spaces and T B(H, K). The following statements are equivalent. 1 T N (H, K), 2 T N (K, H), 3 T N (H, K), 4 T N (K, H), 5 T 2 N (H, K), 6 T 2 N (K, H), When Tx 0 = T, we know T (T 1 T x 0) = T = T.

Properties for N -operators The following are proved by Carvajal and Neves (2012): Theorem Let H and K be separable infinite dimensional Hilbert spaces and T B(H, K). The following statements are equivalent. 1 T N (H, K), 2 T N (K, H), 3 T N (H, K), 4 T N (K, H), 5 T 2 N (H, K), 6 T 2 N (K, H), When Tx 0 = T, we know T (T 1 T x 0) = T = T. When H = K, we know that N(H)(= N (H, H)) is self-adjoint set.

Examples related to N -operators 1 D : l 2 l 2 by D(x 1, x 2,..., x n,... ) = (x 1, x 2 2,..., x n n,... ) for x = (x n ) l 2. Then D 0 and D satisfies the N -property. Moreover, any compact operator satisfies the N -property. 2 Let T = 1 I D. Then 2 T = sup{ 1 2 1 n : n N} = 1 2 = Te 1. Hence T N (H).

Examples related to N -operators 1 D : l 2 l 2 by D(x 1, x 2,..., x n,... ) = (x 1, x 2 2,..., x n n,... ) for x = (x n ) l 2. Then D 0 and D satisfies the N -property. Moreover, any compact operator satisfies the N -property. 2 Let T = 1 I D. Then 2 T = sup{ 1 2 1 n : n N} = 1 2 = Te 1. Hence T N (H). Let M = [e 1 ]. Then M is a reducing subspace of T and T M = sup{ 1 2 1 n : n 2} = 1 2. But T M / N (M, l 2 ), because that neigher 1 2 nor 1 2 is an eigenvalue of T M.

Proposition Let H be a separable, infinite dimensional Hilbert space. Then N (H) is dense in B(H) with respect to the normed topology.

AN -operators Definition Let T B(H, K). Then T is said to be an AN operator or to satisfy the absolutely norm attaining property AN if for every nonzero closed subspace M of H T M : M K satisfies the norm attaining property.

AN -operators Definition Let T B(H, K). Then T is said to be an AN operator or to satisfy the absolutely norm attaining property AN if for every nonzero closed subspace M of H T M : M K satisfies the norm attaining property. When we describe AN (H) as the set of all AN -operators, then we have AN (H) N (H). Moreover, we know that AN (H) N (H).

Necessary conditions for positive AN -operators Lemma (Pandey-Paulsen 2016) Let H and K be separable, infinite dimensional Hilbert spaces. For a closed linear space M of H let V M : M H be an inclusion map from M to H. Then T AN (H, K) if and only if for every nontrivial closed subspace M of H TV M N (M, K).

Necessary conditions for positive AN -operators Lemma (Pandey-Paulsen 2016) Let H and K be separable, infinite dimensional Hilbert spaces. For a closed linear space M of H let V M : M H be an inclusion map from M to H. Then T AN (H, K) if and only if for every nontrivial closed subspace M of H TV M N (M, K). Example Let V be an isometry. Then V AN (H).

Necessary conditions for positive AN -operators Lemma (Pandey-Paulsen 2016) Let H and K be separable, infinite dimensional Hilbert spaces. For a closed linear space M of H let V M : M H be an inclusion map from M to H. Then T AN (H, K) if and only if for every nontrivial closed subspace M of H TV M N (M, K). Example Let V be an isometry. Then V AN (H). Theorem (Pandey-Paulsen 2016) Let T AN be positive. Then H has an orthonormal basis consisting of eigenvelues of T.

Proposition (Pandey-Paulsen 2016) Let T AN (H) be positive. Then the spectrum σ(t ) of T has at most one limit point. Moreover, the uniqune limit point (if exists) can only be the limit of a decreasing sequence in the spectrum. Proof. (Scketch) (Step 1) If there exists an increasing ssequence of {λ n } such that λ λ with λ n < λ (n N). Set M 0 = clos(spn{v n }), where v n s are the eigenvector corresponding to λ n. Then T M0 = sup{ λ n } = λ. However, for every x = n α nv n M 0 with n α n 2 = 1 so that x = 1, T M0 (x) = n α n 2 λ n 2 < λ 2 n α n 2 = λ 2, so that T M0 (x) < λ T M0, contradiction.

Continuation of the proof Proposition (Pandey-Paulsen 2016) Let T AN (H) be positive. Then the spectrum σ(t ) of T has at most one limit point. Moreover, the uniqune limit point (if exists) can only be the limit of a decreasing sequence in the spectrum. (Step 2) Suppose that the spectrum of σ(t ) = clos({β α } α Λ ) has two limits a < b. Then from the step 1 there are decreasing sequences {a n }, {b n } {β α } α Λ such that a n a and b n b. May assume that a 1 < b. Hence a n < b n for n N. Definie M = clos(span{c n f n + 1 c 2 ng n : n N}), where c n [0, 1], Tf n = a n f n and Tg n = b n g n for each n N. Then we know that T M / N (M, H).

Theorem (Pandey-Paulsen 2016) If T AN (H) is positive, then T = α Λ β αv α v α, where {v α : α Λ} is an orthogonal basis consisiting of eigenvalues of T and for every α Λ Tv α = β α v α with β α 0 such that 1 for every finite subset Γ Λ we have sup{β α : α Γ} = max{β α : α Γ}, 2 σ(t ) has at most one limit point. Moreover, the unique limit point (if it exists) can only the limit of a decreasing sequence in σ(t ), 3 the set {β α } α Λ eigenvalues of T, without counting multiplicities, is countable and has at most one eigenvalue with infinite multiplicity, 4 if σ(t ) has both, a limit point β and an eigenvalues ˆβ with infinite multiplicity, then β = ˆβ.

Theorem (Pandey-Paulsen 2016) If T AN (H) is positive, then T = α Λ β αv α v α, where {v α : α Λ} is an orthogonal basis consisiting of eigenvalues of T and for every α Λ Tv α = β α v α with β α 0 such that 1 for every finite subset Γ Λ we have sup{β α : α Γ} = max{β α : α Γ}, 2 σ(t ) has at most one limit point. Moreover, the unique limit point (if it exists) can only the limit of a decreasing sequence in σ(t ), 3 the set {β α } α Λ eigenvalues of T, without counting multiplicities, is countable and has at most one eigenvalue with infinite multiplicity, 4 if σ(t ) has both, a limit point β and an eigenvalues ˆβ with infinite multiplicity, then β = ˆβ. Note that if V : l 2 l 2 be an isometry such that dim V (H) =, then V AN (H), but V / AN (H), and AN (H) N (H).

Properties for AN (H) Theorem (Pandey-Paulsen 2016) Let H be an arbitrary dimensional Hilbert space and T B(H) be a positive. Then T AN (H) if and only if T = αi + K + F, where α 0, K is positive compact operator and F is self-adjoint finite-rank operator.

Properties for AN (H) Theorem (Pandey-Paulsen 2016) Let H be an arbitrary dimensional Hilbert space and T B(H) be a positive. Then T AN (H) if and only if T = αi + K + F, where α 0, K is positive compact operator and F is self-adjoint finite-rank operator. Proposition (Ramesh-Naidu 2017) Let T B(H, K). Then T AN (H, K) if and only if T T AN (H).

Properties for AN (H) Theorem (Pandey-Paulsen 2016) Let H be an arbitrary dimensional Hilbert space and T B(H) be a positive. Then T AN (H) if and only if T = αi + K + F, where α 0, K is positive compact operator and F is self-adjoint finite-rank operator. Proposition (Ramesh-Naidu 2017) Let T B(H, K). Then T AN (H, K) if and only if T T AN (H). Corollary (Ramesh-Naidu 2017) Let T B(H) be normal. Then T AN (H) if and only if T AN (H)

Maps preserving AN -operators Proposition (Osaka-Ramesh 2018) Let V B(H) be an isometry and ϕ: B(H) B(H) be defined by ϕ(t ) = V TV, for T B(H). Then ϕ is a unital C.P. map and ϕ(an (H) + ) AN (H) +. Furthermore, if V AN (H), then ϕ(an (H)) AN (H).

Maps preserving AN -operators Proposition (Osaka-Ramesh 2018) Let V B(H) be an isometry and ϕ: B(H) B(H) be defined by ϕ(t ) = V TV, for T B(H). Then ϕ is a unital C.P. map and ϕ(an (H) + ) AN (H) +. Furthermore, if V AN (H), then ϕ(an (H)) AN (H). Lemma (Carvajal-Neves 2011) Let P B(H) be an orthogonal projection. Then P AN (H) if and only if dim ker(p) < or dim P(H) <.

Maps preserving AN -operators Proposition (Osaka-Ramesh 2018) Let V B(H) be an isometry and ϕ: B(H) B(H) be defined by ϕ(t ) = V TV, for T B(H). Then ϕ is a unital C.P. map and ϕ(an (H) + ) AN (H) +. Furthermore, if V AN (H), then ϕ(an (H)) AN (H). Lemma (Carvajal-Neves 2011) Let P B(H) be an orthogonal projection. Then P AN (H) if and only if dim ker(p) < or dim P(H) <. Corollary (Osaka-Ramesh 2018) Let V B(H) be an isometry and ϕ: B(H) B(H) be defined by ϕ(t ) = V TV, for T B(H). If V / AN (H), then there exists an unitary U such that ϕ(u) AN (H).

Theorem (Osaka-Ramesh 2018) Let A AN (H). Dfinie ρ: B(H) B(H) by ρ(t ) = A TA for T B(H). Then ρ(an (H) + ) AN (H) +. Moreover, if A AN (H), then ρ(an (H)) AN (H).

Theorem (Osaka-Ramesh 2018) Let A AN (H). Dfinie ρ: B(H) B(H) by ρ(t ) = A TA for T B(H). Then ρ(an (H) + ) AN (H) +. Moreover, if A AN (H), then ρ(an (H)) AN (H). Since A AN (H) if and only if A A AN (H), we know that A A = αi + K + F for some α 0, K K(H) +, F = F F (H). Hence, ρ(an (H) + ) AN (H) +.

Theorem (Osaka-Ramesh 2018) Let A AN (H). Dfinie ρ: B(H) B(H) by ρ(t ) = A TA for T B(H). Then ρ(an (H) + ) AN (H) +. Moreover, if A AN (H), then ρ(an (H)) AN (H). Since A AN (H) if and only if A A AN (H), we know that A A = αi + K + F for some α 0, K K(H) +, F = F F (H). Hence, ρ(an (H) + ) AN (H) +. Using this idea we can prove Corollary Let S, T AN (H). Then ST AN (H).

Theorem (Osaka-Ramesh 2018) Let A AN (H). Dfinie ρ: B(H) B(H) by ρ(t ) = A TA for T B(H). Then ρ(an (H) + ) AN (H) +. Moreover, if A AN (H), then ρ(an (H)) AN (H). Since A AN (H) if and only if A A AN (H), we know that A A = αi + K + F for some α 0, K K(H) +, F = F F (H). Hence, ρ(an (H) + ) AN (H) +. Using this idea we can prove Corollary Let S, T AN (H). Then ST AN (H). Question How to characterize the map of the form ρ() = A ()A for some A AN (H)?

Let k be a positive integer. We denote by B k (H) and B k (H), the set of all bounded linear operators of rank k and the set of all bounded linear operators of rank at most k, respectively. We say ϕ : B(H) B(H) is a rank k-preserver if A B k (H) implies ϕ(a) B k (H). We say ϕ is a rank-k non increasing map if A B k (H) implies ϕ(a) B k (H).

Let k be a positive integer. We denote by B k (H) and B k (H), the set of all bounded linear operators of rank k and the set of all bounded linear operators of rank at most k, respectively. We say ϕ : B(H) B(H) is a rank k-preserver if A B k (H) implies ϕ(a) B k (H). We say ϕ is a rank-k non increasing map if A B k (H) implies ϕ(a) B k (H). Theorem (M. Gyory, L. Molnar and P. Semrl 1998) Let k be a positive integer and H be a Hilbert space. Assume that ϕ : B(H) B(H) be a linear rank k non increasing map which is weakly continuous on norm bounded sets. Then either the image of ϕ is a linear space consisting of operators of rank at most k or there exists A, B B(H) such that either ϕ(t ) = ATB for all T B(H) or ϕ(t ) = AT tr B for all T B(H). Here T tr denotes the transpose of T relative to any orthonormal basis of H fixed in advance.

Theorem (O-Ramesh 2018) Let ϕ : B(H) B(H) be a linear positive unital, rank k-non increasing map, which is weakly continuous on norm bounded sets. Assume that ϕ preserves the set AN (H) +. Then there exist an isometry A B(H) such that either ϕ(t ) = A TA for all T B(H) or ϕ(t ) = A T tr A for all T B(H). In addition if ϕ preserves the set AN (H), then A AN (H).

Theorem (O-Ramesh 2018) Let ϕ : B(H) B(H) be a linear positive unital, rank k-non increasing map, which is weakly continuous on norm bounded sets. Assume that ϕ preserves the set AN (H) +. Then there exist an isometry A B(H) such that either ϕ(t ) = A TA for all T B(H) or ϕ(t ) = A T tr A for all T B(H). In addition if ϕ preserves the set AN (H), then A AN (H). Theorem (O-Ramesh 2018) Let ϕ : B(H) B(H) be a linear positive, rank k-non increasing map, which is weakly continuous on norm bounded sets. Assume that ϕ preserves the set AN (H) + and ϕ(i ) invertible. Then there exist A AN (H) such that either ϕ(t ) = A TA for all T B(H) or ϕ(t ) = A T tr A for all T B(H). In addition if ϕ preserves the set AN (H) and ϕ(i ) 1 AN (H), then A AN (H).

Dropping the invertibility of ϕ(i ), to some extent we can generalize the result in the previous Theorem.

Dropping the invertibility of ϕ(i ), to some extent we can generalize the result in the previous Theorem. Theorem (O-Ramesh 2018) Let ϕ : B(H) B(H) be a linear positive, rank k-non increasing map, which is weakly continuous on norm bounded sets. Assume that ϕ preserves the set AN (H) +. Further assume that there exists S B(H) such that rank of ϕ(s) is not finite. Then there exist A AN (H) such that either ϕ(t ) = A TA for all T B(H) or ϕ(t ) = A T tr A for all T B(H).

References 1 X. Carvajal and W. Neves, Operators that achieve the norm, Integral Equations Operator Theory 72 (2012), no. 2, 179 195. MR2872473 (2012k:47044) 2 C. K. Li and S. Pierce, Linear preserver problems, Amer. Math. Monthly 108 (2001), 591 605. MR1862098 3 L. Molnar, Selected preserver problems on algebraic structures of linear operators and on function spaces, Lecture Notes in Mathematics, 1895, Springer-Verlag, Berlin, 2007. MR2267033 4 H. Osaka and G. Ramesh, Maps preserving AN -operators, preprint 2018. 5 V. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, 78, Cambridge University Press, Cambridge, 2002. MR1976867 6 Satish Pandey and Vern Paulsen, A Spectral Characterization of AN - Operators, Journal of the Australian Mathematical Society, 2016, doi 10.1017/S1446788716000239 7 G. Ramesh, Structure theorem for AN -operators, J. Aust. Math. Soc. 96 (2014), no. 3, 386 395. MR3217722 8 G. Ramesh and D. Venkunaidu, On Absolutely norm attaining operators, preprint 2016, arxiv:1801.02432. 9 W. F. Stinespring, Positive functions on C -algebras, Proc. Amer. Math. Soc. 6 (1955), 211 216. MR0069403

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