Banach Journal of Mathematical Analysis ISSN: (electronic)

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1 Banach J. Math. Anal. 6 (2012), no. 1, Banach Journal of Mathematical Analysis ISSN: (electronic) AN EXTENSION OF KY FAN S DOMINANCE THEOREM RAHIM ALIZADEH 1 AND MOHAMMAD B. ASADI 2 Communicated by F. Kittaneh Abstract. We prove that for a separable Hilbert space H with an orthonormal basis {e i }, the equality = s i( )e i e i holds for all unitarily invariant norms on B(H) and Ky Fan s dominance theorem remains valid on B(H). 1. Introduction There has been a great interest in studying unitarily invariant norms and symmetric norm ideals on B(H) in the last few decades (see, e.g., [1]-[4],[5],[7],[9]-[12] and the references therein). A norm on a non-zero ideal J of B(H) is called unitarily invariant if UT V = T for all unitary operators U, V B(H) and T J. The ith s-number of an operator T on H is displayed by s i (T ) and is given by s i (T ) = inf{ T F op : F B(H) has rank < i}, where op denotes the usual operator norm on B(H). Note that every finite rank operator belongs to any non-zero ideal of B(H). Typical examples of unitarily invariant norms on B(H) are Ky Fan k-norms that are defined by N k ( ) = s 1 ( ) + + s k ( ) [3], see also [10]. We say that a norm on J, satisfies Ky Fan s dominance theorem, if for every T, R J, with N k (T ) N k (R) for all k N, the inequality T R holds. Ky Fan s dominance theorem holds for J if it holds for all unitarily invariant norms on J. Date: Received: 26 July 2011; Accepted: 9 November Corresponding author Mathematics Subject Classification. Primary 47A05; Secondary 47A30. Key words and phrases. s-numbers, unitarily invariant norm, Ky Fan norm. 139

2 140 R. ALIZADEH, M.B. ASADI In the finite dimensional case, owing to the presence of the singular value decomposition (SVD), there is a nice representation of unitarily invariant norms as symmetric gauge functions [2, Theorem ], which plays a major role in solving problems and proving theorems in the finite dimensional case. In fact using SVD, we conclude that for every matrix A M n (C) the equality A = diag(s 1 (A),, s n (A)) is satisfied for all unitarily invariant norms on M n (C), where M n (C) is the algebra of all n n matrices with the entries in C. In this paper we prove an alternative equality in the infinite dimensional case. In fact, we show if H is a separable Hilbert space with an orthonormal basis {e i }, the equality = s i( )e i e i holds for all unitarily invariant norms on B(H), where (s i ( )e i e i )(h) = s i ( ) < h, e i > e i, for all h H. As a corollary we conclude that for a separable Hilbert space H, Ky Fan s dominance theorem remains valid on B(H). 2. Unitarily invariant norms on B(H) Thought this section, when we say J is a non-zero ideal of B(H), it is possible for J to be equal the whole B(H). Also will be an arbitrary unitarily invariant norm on B(H). We say that a norm on J is symmetric if T 1 ST 2 T 1 op S T 2 op for all T 1, T 2 B(H) and S J. Lemma 2.1. Every unitarily invariant norm on a non-zero ideal J of B(H) is symmetric. Proof. Let T B(H), S J and consider a real number α > 1. By [8], there are T unitary elements U 1,, U n in B(H) such that α T op = U 1+ +U n. Hence n T T S = α T op S α T op = α T U U n op S n α T op S. Since α > 1 is arbitrary, the inequality T S T op S holds. Similarly we can show that ST S T op. Corollary 2.2. Let be a unitarily invariant norm on a non-zero ideal J of B(H) and T, S J. Then (i) T = T. (ii) T = T. (iii) p = q, for any equivalent projections p and q in J. (iv) If T S 0, then T S. Proof. The polar decomposition T = u T of T implies that T = u T, T = T u and T = T u. Also, if p and q are equivalent then p = vv and q = v v, for some partial isometry v in B(H) and hence we have v pv = q and vqv = p. If T S 0, there is an operator R with R op 1 such that S = RT. These arguments together with Lemma 2.1 imply (i)-(iv). Lemma 2.3. Let J be a non-zero ideal of B(H), T J and P be a projection of rank one. For every unitarily invariant norm on J the inequality T P T op holds.

3 AN EXTENSION OF KY FAN S DOMINANCE THEOREM 141 Proof. Suppose T 0 and consider a sequence {x n } n=1 in H such that x n = 1, for all n and lim n T (x n ) = T op. Without loss of generality we can suppose that T (x n ) 0, for all n N. Let U n be a unitary operator that U n ( T (xn) ) = x T (x n) n. Setting P n = x n x n we have T = P n op T T P n = T (x n ) T (x n ) x n T (x n) ( ) = T U (xn ) n T (x n ) x n T (x n ) = x n x n T (x n ) = P n T (x n ) = P T (x n ), where the last equality has resulted from (iii) of Corollary 2.2. Now if n we get the desired result. Corollary 2.4. If P is a projection of rank one in B(H) then P T op T I T op (T B(H)), where I is the identity operator on H. Therefore, all unitarily invariant norms on B(H) are equivalent to the operator norm. Corollary 2.5. If P is a projection of rank one in B(H) and P = I, then is a multiple of the operator norm. Lemma 2.6. Suppose {e i } is an orthonormal sequence in H. For positive diagonal operator T = λ ie i e i (λ i 0), let E = {λ i i = 1, 2, }. Then (i) s 1 (T ) = T op = sup j N λ j. (ii) If there exist k 1 distinct positive integers n 1,, n k 1 such that s i (T ) = λ ni (1 i k 1), then s k (T ) = sup j / {n1,,n k 1 }λ j. Also, if s k (T ) is a limit point of E, then for every i N, we have s k+i (T ) = s k (T ). (iii) If there is no s-number of T that is a limit point of E, then there are distinct positive integers n 1, n 2,... such that s i (T ) = λ ni, i N. Otherwise there is positive integer k and k 1 distinct natural numbers n 1,..., n k 1 such that s i (T ) = λ ni, 1 i k 1 and s k (T ) = s k+1 (T ) =. In both cases, for every positive integer i, we have s i (T ) = sup λj s i (T )λ j. Proof. (i) This is a well known fact [6, Problem 63]. (ii) If k = 1 the equality is resulted from (i). Otherwise setting we have rank(f ) < k and so k 1 F = λ ni e ni e ni, s k (T ) T F op = sup j / {n1,,n k 1 }λ j.

4 142 R. ALIZADEH, M.B. ASADI On the other hand if for every j N \ {n 1,, n k 1 }, setting k 1 R j = λ ni e ni e ni + λ j e j e j, we have R j T. Since Ky Fan norms are unitarily invariant, using (iv) of Corollary 2.2, we have N k (R j ) N k (T ). Therefore λ j s k (T ) and hence s k (T ) = sup j / {n1,,n k 1 }λ j. If s k (T ) is a limit point of E, then for every ɛ > 0 and i N, there exist distinct positive integers m 1,, m i+1 N \ {n 1,, n k 1 } such that s k (T ) ɛ i + 1 < λ m j s k (T ), (1 j i + 1). Setting i+1 k 1 S = λ mj e mj e mj + λ nj e nj e nj, we have 0 S T and so N k+i (S) N k+i (T ). This implies that i+1 (s k (T ) ɛ i + 1 ) s k(t ) + + s k+i (T ) s k (T ) + + s k (T ) +s }{{} k+i (T ). i times Therefore s k (T ) ɛ s k+i (T ) and so s k (T ) s k+i (T ). Hence, s k (T ) = s k+i (T ). (iii) By (i) we have s 1 (T ) = sup j N λ j. If s 1 (T ) is a limit point of E, then by the second part of (ii), we have s 1 (T ) = s 2 (T ) =. Otherwise there is n 1 N such that s 1 (T ) = λ n1 and by the first part of (ii) we have s 2 (T ) = sup j / {n1 }λ j. Now if s 2 (T ) is a limit point of E, then again by the second part of (ii), we have s 2 (T ) = s 3 (T ) =. Otherwise there is n 2 N {n 1 } such that s 2 (T ) = λ n2 and by the first part of (ii), we have s 3 (T ) = sup j / {n1,n 2 }λ j. Continuing this process we get desired results. In particular, the following corollary follows from the previous lemma. Corollary 2.7. Suppose {e i } is an orthonormal sequence in H. For positive diagonal operator T = λ ie i e i, let s = inf{s i (T ) i N}, E = {λ i i = 1, 2, }. Then (i) for every ɛ > 0, there exist distinct positive integers n 1, n 2, such that 0 s i (T ) < λ ni + ɛ. (ii) for every ɛ > 0, A = {i λ i > s + ɛ} is a finite set. In fact, A is empty or there exist distinct positive integers n 1, n N0, such that A = {n i : 1 i N 0 } and λ ni = s i (T ) (1 i N 0 ). Lemma 2.8. Suppose {e i } is an orthonormal sequence in H and T = λ ie i e i (λ i 0) is a positive diagonal operator in B(H). Then

5 AN EXTENSION OF KY FAN S DOMINANCE THEOREM 143 (i) for every ɛ > 0 there exists a unitary element U B(H) such that UT U (s i (T ) + ɛ)e i e i, (ii) for every ɛ > 0 there exists a partial isometry U B(H) such that (s i (T ) ɛ)e i e i UT U, (iii) T = s i(t )e i e i. Proof. Let ɛ > 0, s = inf{s i (T ) i N} and A = {i λ i > s + ɛ}. The previous corollary implies that A is empty or there exist distinct positive integers n 1, n N0, such that A = {n i : 1 i N 0 } and λ ni = s i (T ) (1 i N 0 ). If A is empty, we set U = I, otherwise we consider U as a unitary operator that maps {e n } n=1 onto {e n } n=1 and U(e ni ) = e i for i = 1,, N 0. Therefore N 0 UT U = s i (T )e i e i + µ i e i e i, i=n 0 +1 where µ i {λ j : j n i, for all 1 i N 0 }. Since for all i N 0 we have µ i s + ɛ, then UT U (s i(t ) + ɛ)e i e i. For proving (ii), we recall that there exist distinct positive integers n 1, n 2, such that 0 s i (T ) < λ ni + ɛ. Now consider a partial isometry U which satisfies { ei j = n U(e j ) = i, for some i N, 0 otherwise. We have UT U = λ ni e i e i (s i (T ) ɛ)e i e i. Finally (iii) follows from (i),(ii) and (iv) of Corollary 2.2. Lemma 2.9. Let H be a separable Hilbert space and T be a positive operator in B(H). For every ɛ > 0, there exists a diagonal operator T ɛ such that s i (T ) s i (T ɛ ) < ɛ, for all i N. Proof. By [11] there exist a diagonal operator T ɛ and a compact operator K ɛ such that T = T ɛ + K ɛ and the Hilbert-Schmidt norm of K ɛ is less than ɛ. Hence for every finite rank operator F, the following inequalities hold T ɛ F op ɛ T F op T ɛ F op + ɛ. Taking infimum over F with rank(f ) < i, we get the desired result. Theorem Let H be a separable Hilbert space and T B(H). The equality T = s i(t )e i e i holds, for all orthonormal sequence {e i } i N in H.

6 144 R. ALIZADEH, M.B. ASADI Proof. Since T = T and s i (T ) = s i ( T ), we can suppose that T is a positive operator. Let ɛ > 0 and T ɛ = λ ie i e i be the diagonal operator of the previous lemma. We have by Lemmas 2.1 and 2.8 T T ɛ + K ɛ T ɛ + I K ɛ op s i (T ɛ )e i e i + ɛ I s i (T )e i e i + 2ɛ I. Similarly We have s i (T )e i e i s i (T ɛ )e i e i + ɛ I = T ɛ + ɛ I T + 2ɛ I. Lemma Suppose {e i } i N is an orthonormal sequence in H and D 1 = λ ie i e i, D 2 = µ ie i e i (λ i, µ i 0), are positive diagonal operators in B(H). Assume moreover that there exists N N such that λ k = µ k for all k > N and s k (D 1 ) = λ k, s k (D 2 ) = µ k for all 1 k N. If N k (D 1 ) N k (D 2 ) for all 1 k N, then D 1 D 2. Proof. Let X 1 = N λ ie i e i and X 2 = N µ ie i e i. We have k λ i k µ i, for every 1 k N and so, there are unitary matrices U 1,, U 2 N N! in M N (C) and non-negative numbers c 1,, c 2 N N! such that X 1 = 2 N N! c j U j X 2 Uj and 2 N N! c j = 1 [2, II.2.10]. Now, we can choose unitary operators Ũ1,, Ũ2 N N! in B(H) such that N λ i e i e i = 2 N N! ( N ) c j Ũ j µ i e i e i Ũj, and Ũj(e i ) = e i, for every 1 j 2 N N! and i > N. A direct computation shows that D 1 = 2 N N! c j Ũ j D 2 Ũ j, and so D 1 D 2. Now, we can show that Ky Fan s dominance theorem is valid on B(H), where H is a separable Hilbert space. Theorem Let H be a separable Hilbert space and T, R B(H). If N k (T ) N k (R) for all k N, then T R. Proof. For every ɛ > 0, we can choose N N such that: i) s N (T ) s N (R) + ɛ,

7 AN EXTENSION OF KY FAN S DOMINANCE THEOREM 145 ii)s N (R) s i (R) + ɛ, for every i N. Using (iv) of Corollary 2.2 and Lemma 2.10 together with Lemma 2.11, we have T = s i (T )e i e i N s i (T )e i e i + s N (T )e i e i i=n+1 N s i (R)e i e i + s N (T )e i e i i=n+1 N s i (R)e i e i + (s N (R) + ɛ)e i e i i=n+1 N s i (R)e i e i + (s i (R) + 2ɛ)e i e i i=n+1 s i (R)e i e i + 2ɛ I = R + 2ɛ I. Acknowledgment. The research of the second author was in part supported by a grant from IPM (No ). References 1. J. Alaminos, J. Extremera and A.R. Villena, Uniqueness of rotation invariant norms, Banach J. Math. Anal. 3 (2009), no. 1, R. Bhatia, Matrix Analysis, Springer-Verlag, New York, J.T. Chan, C.K. Li and C.N. Tu, A class of unitarily invariant norms on B(H), Proc. Amer. Math. Soc. 129 (2000), no. 4, J. Fang, D. Hadwin, E. Nordgren and J. Shen, Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property, J. Funct. Anal. 255 (2008), no. 1, I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Transl. Math. Monog., 18, Amer. Math. Soc., Provid., P.R. Halmos, A Hilbert Space Problem Book, Springer-Verlag, Berlin, R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, R.V. Kadison and G.K. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), no. 2, C.K. Li, Some aspects of the theory of norms, Linear Algebra Appl. 219 (1995), M. S. Moslehian, Ky Fan inequalities, Linear Multilinear Algebra (to appear), Available online at D. Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators, J. Operator Theory 2 (1979), J. Von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937),

8 146 R. ALIZADEH, M.B. ASADI 1 Department of Mathematics, Shahed University, P.O. Box , Tehran, Iran. address: alizadeh@shahed.ac.ir 2 School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Enghelab Avenue, Tehran, Iran; School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box: , Tehran, Iran. address: mb.asadi@khayam.ut.ac.ir