111
Faculty of Sciences and Mathematics
University of Niš
Available at:
www.pmf.ni.ac.yu/sajt/publikacije/publikacije_pocetna.html
Filomat 21:1 (2007), 99 – 111______
SOME SINGULAR VALUES, AND UNITARILY INVARIANT
NORM INEQUALITIES CONCERNING
GENERALIZED INVERSES
P. J. Maher
Abstract:
We sharpen and extend inequalities concerning generalized inverses previously obtained for the von Neumann-Schatten, and supremum, norms. We sharpen those inequalities to obtain corresponding inequalities for singular values for ; and we extend those inequalities, for finite rank operators, to inequalities for an arbitrary unitarily invariant norm.
Received: December 24, 2006
Keywords and phrases: Generalized inverses, Moore-Penrose inverse, von Neumann-Schatten classes and norms.
2000 Mathematics Subject Classification: 47A05, 47B10.
1. INTRODUCTION
This paper sharpens and extends inequalities concerning generalized inverses previously obtained for the von Neumann-Schatten, and supremum, norms. It sharpens those inequalities to obtain corresponding inequalities for singular values and it extends them, at least for finite rank operators, to inequalities concerning an arbitrary unitarily invariant norm. For example, as is well-known [5, Theorems 2.1, 2.2], [6, Theorem 3.3], if A has a (i), (iii) inverse (meaning that if A satisfies and ) and if , where , then
(1.1)
(where denotes the von Neumann-Schatten class and its norm). Here, in Corollary 3.1 below, we prove that
for (1.2)
where the singular values and are – as always – arranged in decreasing order and repeated according to multiplicity; and we prove that if is of finite rank then so is and for every unitarily invariant norm
. (1.3)
The finite rank condition is required – as far as unitarily invariant norms are concerned – so that we can appeal to the von Neumann-Schatten theory of unitarily invariant norms and their representation by symmetric gauge functions [8, Chapter 10], [12, Chapter V].
The singular values inequality (1.2) is deduced as a corollary to the main result of this paper, Theorem 3.1: this shows that a modulus inequality between operators implies one between singular values which, for finite rank operators, implies an inequality between unitarily invariant norms (just as in [6, Lemma 3.1] an inequality between moduli of operators implies one between von Neumann-Schatten norms). Another consequence of Theorem 3.1 is Theorem 3.2: this says that if , where are fixed and P and Q are projections, and if for then, amongst other things, for . Theorem 3.2 yields as corollaries (Corollaries 3.2, 3.3 and 3.4) a host of sharpenings/extensions of already known results to do with generalized inverses including the result, Corollary 3.4 [5, Theorems 3.1 and 3.2] that for , where is a generalized inverse of A and is the Moore-Penrose inverse of A.
Theorem 3.2 and Corollaries 3.2, 3.3 and 3.4 are, as pointed out below, essentially finite dimensional since their hypotheses imply that the operator A occurring in them (and its adjoint ) must be of finite rank.
For unitarily invariant norms (not necessarily on operators of finite rank) there is a property of strict convexity, analogous to the strict convexity of for ; and, as with , strict convexity implies a uniqueness property (stated in Lemma 2.3). Thus, in (3.3), for example, equality holds if, and for strictly convex only if, .
2. PRELIMINARIES
Throughout, H is a complex, separable Hilbert space with inner product and norm ; and all operators are in , the space of all bounded, linear operators mapping H to H. An operator A is self-adjoint if ; equivalently, if for all f in H; a (self-adjoint) operator is positive, denoted , if for all f in H. The notation , for self-adjoint A and B, means that . It can be shown that every positive operator T has a unique positive square root, denoted . The modulus of an arbitrary operator A is the positive square root of (the positive operator) ; that is, .
Recall that an operator is a generalized inverse of A if . An operator A has a generalized inverse if and only if its range, , is closed (We define [13, p.251, Theorem 12.9]). For an operator A, with closed range, its Moore-Penrose inverse satisfies
/ (i) / (2.1)/ (ii)
/ (iii)
/ (iv) ;
and, further, is uniquely determined by these properties [9, Theorem 1]. For the construction of the Moore-Penrose inverse of an operator with closed range see [13, p.251, Theorem 12.9]. If an operator satisfies (i) and (iii) of (2.1) (so that and ) it will be called a (i), (iii) inverse of A; if satisfies (i), (iv) of (2.1) it will be called a (i), (iv) inverse of A. Observe that if is a (i), (iii) inverse of A then is the projection onto and that if is a (i), (iv) inverse of A then is the projection onto .
An operator A is of finite rank n, denoted , if . The rank 1 operator , for fixed vectors f and g in H, is denoted . The spectral theorem for compact, positive operators says that every compact, positive operator X can be expressed uniquely as
(2.2)
where is the sequence of positive eigenvalues of X arranged in decreasing order and repeated according to multiplicity, and is the corresponding orthonormal sequence of eigenvectors (so that ); and X is of finite rank n if and only if the sequence terminates after just n terms [11, p.64 (2), p.70 cf. Theorem 1.9.3]. In particular, if , for some A in , the eigenvalues are called the singular values of A and denoted .
The spectral theorem for compact operators says that every compact operator X can be expressed uniquely as
(2.3)
where and are orthonormal sequences in H and is the sequence of singular values of X, arranged in decreasing order and repeated according to multiplicity; and X is of finite rank n if and only if the sequence terminates after just n terms [11, Theorem 1.9.3].
For a compact operator A, let , , … denote the singular values of A, arranged in decreasing order and repeated according to multiplicity. If, for some , we say A is in the von Neumann-Schatten class and write
.
For , it can be shown that is a norm called the von Neumann-Schatten norm. We sometimes write , the supremum norm. For all p, where , is a 2-sided ideal of and for the space is a Banach space under . For more details of the von Neumann-Schatten classes and norms see [2, Chapter XI], [11, Chapter 2].
The class is called the trace class. If and is an orthonormal basis of H then the quantity , called the trace of A and defined by , can be shown to be finite and independent of the particular basis chosen. Further, , where , if and only if .
From (2.3) it follows that for every X in , where , there exists a sequence of operators, each of finite rank n, such that
as . (2.4)
The von Neumann-Schatten norms , for , satisfy the property of uniformity [2, Chap XI, 9.9(d)]: if , and then
. (2.5)
We cite three further results we shall need about the von Neumann-Schatten norms.
LEMMA 2.1 [6, Lemma 3.1]. (a) If then
(b) if, further, A is compact then is compact and
(c) if, further, , for , then and
Recall that the polar decomposition [3, Chapter 16] says that every operator A can be expressed uniquely as where the partial isometry U is such that (A partial isometry U satisfies for all f in ).
THEOREM 2.1 [1, Theorem 2.1]. If , the map (from to ) is differentiable with derivative at X given by
where is the polar decomposition of X. If the underlying space H is finite-dimensional the same result holds for at every invertible element X.
LEMMA 2.2 [6, Lemma 2.5]. If S is a convex set of operators in , where , there is at most one minimizer of where .
A norm is said to be unitarily invariant if for all unitary operators U and V (provided ). Examples of unitarily invariant norms are the supremum norm and the von Neumann-Schatten norms , .
A unitarily invariant norm is strictly convex if
for real, positive k (provided and ). The next result – a generalization of Lemma 2.2 – is proved similarly to it.
LEMMA 2.3. Let be a strictly convex, unitarily invariant norm and let , say. Then if S is a convex set of operators in there is at most one minimizer of where .
Many properties of unitarily invariant norms can be deduced via Theorem 2.2 from those of symmetric gauge functions provided the operators concerned are of finite rank. References below are to Schatten’s own elegant exposition [12].
Let F be the set of all operators of finite rank and let L be the set of all sequences of real numbers having a finite number of non-zero terms. A symmetric gauge function is a function satisfying the following properties:
if / (i) / (2.6)/ (ii)
/ (iii)
/ (iv)
where each and is a permutation of . (In (2.6) (iv), and below, we write instead of for the values of the symmetric gauge function .)
LEMMA 2.4 [12, p.68, cf. p.61, Lemma 6]. Let be a symmetric gauge function. If for then
THEOREM 2.2 [12, p.69, Theorem 8]. Let A be in F and let be the singular values of A arranged in decreasing order and repeated according to multiplicity. If is a symmetric gauge function then the function
is a unitarily invariant norm , i.e., ; and, conversely, if is a unitarily invariant norm then there exists a symmetric gauge function such that
.
It follows from Theorem 2.2, first, that every unitarily invariant norm is self-adjoint, i.e., (because the non-zero eigenvalues of and are equal); and, second, that every unitarily invariant norm has the property of uniformity: if , and then [12, p.71, Theorem 11]
. (2.7)
3. SINGULAR VALUES, AND NORM, INEQUALITIES
THEOREM 3.1. Let A be in , where , and B such that . Then:
(a) and
for
(b) if then and for every unitarily invariant norm
Proof. (a) Special case. Suppose, first, that . Then and, as a consequence of (2.2),
where , and (= ). As is compact and it follows from Lemma 2.1 (b) that is compact. Therefore,
where , and (=. We prove that ; equivalently, . Let f be in . Then, since by Lemma 2.1 (b), we have
so that . Hence, . Therefore, and so .
Since and are of finite rank and since for we can apply Lőwner’s result [8, p.510, A1b] that says that if S and T are positive matrices such that then (here, denotes the (positive) eigenvalue of S, the eigenvalues being arranged in decreasing order and repeated according to multiplicity). Hence, , that is, for .
General case. We now extend this result to the von Neumann-Schatten classes. Let now A be in , where . Then, since , it follows from Lemma 2.1 (c), (b) that and . As then . Thus, is a positive operator in and so by (2.2) can be expressed as
where , and (=). Thus, for each fixed n (<) each reduces . Therefore, by the second paragraph of the proof of the special case
for . (3.1)
To extend this to all recall (cf. (2.4)) that the operator X () is the uniform limit in of a sequence of finite rank operators , where . Thus (cf. (2.3))
so that whence, for ,
as and so . Taking X () the inequality (3.1) now extends to all i.
(b) As in the Special case of (a), it follows that if then . If is the symmetric gauge function associated, by Theorem 2.2, with the unitarily invariant norm then, since for , it follows from Lemma 2.4 that
,
that is, ■
COROLLARY 3.1 (cf. [5, Theorems 2.1, 2.2], [6, Theorem 3.3]). Let A have closed range and have a (i), (iii) inverse and let X be such that where . Then
for (3.2);
and if is of finite rank so is and for every unitarily invariant norm
(3.3)
with equality in (3.3) if, and for strictly convex only if, , that is, for arbitrary L in ; if the least such minimizer of finite rank (with respect to ) is .
Proof. As in [6, Theorem 3.3] the inequality yields, by Theorem 3.1 (a), the inequality (3.2) and for finite rank yields, by Theorem 3.1 (b), the inequality (3.3).
Equality holds in (3.3) if , that is, if for arbitrary L. If, further, the inequality of [6, Theorem 3.3] yields, by Theorem 3.1 (b), that is a least such minimizer with respect .
For strictly convex the equality assertions follow from Lemma 2.3 and the convexity of the sets and . ■
There is a similar “left-handed” result pertaining to (cf. [5, (2.4)], [6, (3.4)]). This says, amongst other things, that if B has closed range and has a (i), (iv) inverse and if , where , then
for . (3.4)