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 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 2011. Corresponding author. 2010 Mathematics Subject Classification. Primary 47A05; Secondary 47A30. Key words and phrases. s-numbers, unitarily invariant norm, Ky Fan norm. 139
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 3.5.18], 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 1 + + 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.
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.
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
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 2.10. 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.
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 2.11. 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 2.12. 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,
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146 R. ALIZADEH, M.B. ASADI 1 Department of Mathematics, Shahed University, P.O. Box 18151-159, Tehran, Iran. E-mail 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: 19395-5746, Tehran, Iran. E-mail address: mb.asadi@khayam.ut.ac.ir