Mathematische Annalen

Math. Ann. 281, 7 12 (1988)

(~) Springer-Verlag 1988

On the Clarkson-McCarthy Inequalities Rajendra Bhatia I and John A. R. H o l b r o o k 2'* i Indian Statistical Institute, New Delhi 110016, India 2 University of Guelph, Guelph, Ontario, NIG 2WI, Canada

1. Introduction

C. A. M c C a r t h y proved in [7], a m o n g several other results, the following inequalities for Schatten p-norms of Hilbert space operators: 2(IIA lip + IIBllP)_< ]]A + B]I~+ for 2 ~ p <

]IA--B]I~<2v-~([[A[fv+ HBIIW)

(1)

oo, and 2v-I(IIAIIW+ [IB[[~)-< IIA+BII~+

ItA--BIIW<=2(IIAI[~+[IB[I~)

(2)

for 1 < p < 2 . These are n o n - c o m m u t a t i v e analogues of some inequalities of Clarkson for the classical Banach spaces and constitute one half of the " C l a r k s o n - M c C a r t h y Inequalities." These estimates have been found to be very powerful tools in o p e r a t o r theory and in mathematical physics. (See, e.g., Simon [11].) Here we formulate and prove a more general version of these inequalities. O u r analysis extends these inequalities to a wider class of n o r m s which includes the p-norms and at the same time leads to a p r o o f which is much simpler than M c C a r t h y ' s original p r o o f or some later proofs. Indeed, it appears to be simpler and more elementary than any other p r o o f of which we are aware; see the discussion in [11]. Let ~(~'~) denote the space of all b o u n d e d linear operators on a Hilbert space • . For convenience, we take J f to be infinite-dimensional. If an o p e r a t o r A is compact, we enumerate the eigenvalues of the positive o p e r a t o r (A'A) 1/2 as Sl(A) =>s2(A ) __>.... These are called the singular values of A. An o p e r a t o r A is said to belong to the class J v if ~. j--1

(ss(A))P< ~,

where p is a real number, 1 < p < 0o. If

A r J v then the Schatten p-norm of A is the n u m b e r I[A [Iv = (~(sj(A))V) l/v" It is well known that J v is an ideal in ,~(ouf), that ]lAIIv defines a n o r m on it, and that it is * Supported in part by Indian Statistical Institute, New Delhi, and NSERC of Canada (under operating grant A 8745)

8

R. Bhatia and J. A. R. Holbrook

complete with respect to this norm. See Gohberg and Krein [5], Schatten [10], or [11]. These norms are special examples of symmetric norms or unitarily invariant norms each of which arises as a "symmetric gauge function" of the singular values. (See [10] for definitions.) Each such norm Ill" Ill is defined on a natural subclass Jill. III~ called the norm ideal associated with the norm I]}" [}}and satisfies the invariance property lllUAVt[I = lltAll[ for all A in this ideal and for all unitary operators U, V. The usual operator norm II 9 I[ is also such a norm defined on all of ~ ( ~ ) and, for compact A, IIAII=sl(A). It is hence conventional to denote IIAII by )lal]+. Let 2=
Let R ~ be the space of all sequences of positive real numbers. Given two elements {xj} and {yj} of this space define another element by setting {xj} v {yj} = {XD Yl, X2, Y2. . . . }. Let Itl" III be any unitarily invariant norm on ~ ( ~ ) and let ~b be the associated symmetric gauge function on IRT, i.e., EllAI[{= q~({sj(A)}). Given two operators A and B we define HIAGBIII = q~({sj(A)} v {sj(B)}). This quantity is simply the '[I IH-norm of A O B regarded as the operator ( A in ~ ( ~ @ ~ ) . Note that I[AOBil =max(JlA4{, lIBll),

IIA@BIIp:(HAII~,-4-}IBII~,) x/p for

l
0B)

On the Clarkson-McCarthy Inequalities and in particular

IIA|

9 p for

l
Extensions to direct sums involving more than two operators are obtained in the same way. If the operator ideal J + is normed by the symmetric gauge function q~ then so is J+0).Jt+ by the above procedure. The proof of (1) and (2) in [-7] goes via the following inequalities, which are of independent interest. If A, B are positive operators in 3tp for any p > 1, then 2~-PlIA+ BII~< I[a[l~+ Ilnl[~< [Ih+ BIIg.

(3)

Note that in the notations defined above this can be rewritten as 89

+ B)|

+ B)I], < IIA@Blip < ]I(A+ B)GOIIv.

(4)

Thus the following theorem (its history is outlined in the next section) includes a generalisation of (3). Theorem 1. Let A, B be any two positive operators belonging to the norm ideal associated with a unitarily invariant norm 1{[.Ill. Then

89

< ][IAOB[[[ < [H(A+ B)@01[r.

(5)

To recast (1) and (2) in a similar mould we need to go to quadruplets instead of pairs. Thus, for example, the second inequality in (1) can be rewritten first as 2a/"(IIA+BII~+ IIA-BII~)'/P<2(IIAII~+ IIBII~)'/p and then as II(A+ B)+(A + B)@(A - B ) + ( A - B)llp < 2rl A + 0 + B @ 0 H , for 2 = < p < ~ . Note that the first inequality in (1) can be obtained from the second one by replacing A and B by A + B and A - B respectively, and vice versa. Similar considerations apply to the pair of inequalities in (2). The following two theorems, then, are the promised generalisations of (1) and (2). Theorem 2. Let A and B be two operators belonging to the ideal JQ associated with a Q-norm 11" IIQ. Then

II(A+B)|174

(6)

Theorem 3. Let A and B be two operators belonging to the ideal JQ, associated with a Q*-norm II " I{Q*. Then

2IIA|

< H(A+B)O(A+B)G(A-B)|

(7)

3. Proofs of the Results

In finite dimensions Theorem 1 is a restatement of a majorisation result due to Thompson [12]. One proof of Thompson's result given in Ando [1] goes through, without any change, to infinite dimensions. For the convenience of the reader we reproduce this short and elegant proof.

10

R. Bhatia and J. A. R. Holbrook

To prove the first inequality in (5) write (A + B)| + B)= (A| (BOA) and note that the two terms on the right-hand side have the same norm. To prove the second, write

Then note

(Ao, )xx. whereX (A;/2 ;j2) X ' X = (B~/EAU2 qq .

By

properties

general

of

[][XX*I,[=IIIX*XIIJ a n d ( o

unitarily

invariant

norms

(see,

e.g.,

[5]),

OB),beinga"pinching"ofX*X, hassmallernorm

than X*X. This proves the second inequality in (5). Recently, in [3], we have begun a study of"weakly unitarily invariant" norms. These are norms on spaces of finite-dimensional operators that are invariant under unitary conjugations A ~ U*A U. The pinching inequality extends readily to these norms and, in a finite-dimensional setting, X*X and XX* are unitarily conjugate. Thus the proof above shows that the inequalities (5) are valid for this extended class of norms. Such a norm z' gives rise to a unitarily invariant, norm r by the same procedure as we have used to define Q-norms: z(A)= (z'(A*A)) ~/2 (provided this r satisfies the triangle inequality). On this basis analogues of Theorems 2 and 3 may be formulated and proved in the new setting. We now turn to the proofs of Theorems 2 and 3. Since A*A| and B*B| are positive operators, the first inequality of(5) shows that, for any unitarily invariant

norm II1"I11,

2[II(A*A+B*B)@O|174

<=4111A*A@O|

(8)

invariance and the relation 2(A*A+B*B)=C++C - where C+=(A+B)*(A+B), C-=(A--B)*(A-B), the left side of (8) is III((C+| +)

By unitary

+(C-|174174

By the second inequality of (5) this is not less than Thus we have

ILIC+|174174 IlIC+ |

- III_-<1]I4A* A O O G 4 B * B G O I I L ,

@C- |

for every unitarily invariant norm. Hence, the inequality (6) is true for all Q-norms. We shall obtain (7) from (6), by duality. It is a central result of the Schatten theory (see [10]) that JQ. is the Banach space dual of JQ under the bilinear pairing

(T, S ) = tr TS. We apply this to operators in ~ = ~ -

r,+r3 2

TI+T3

| ~

T,-T3

| ~

~

@~ '

(+) ~f

. For T e ~ let A(T)

where the Tk are the diagonal blocks in

the 4 x 4 operator block matrix corresponding to T. Clearly A is a linear map on ~ and we claim that it is contractive with respect to II 9 lie. To see this note that the

pinching inequality ensures that IITII e > II7"1| Z2| Z3 | T4 IIe, while this is the same as IIT1| - T2| T3| - T4 IIe by unitary invariance. Hence each is no less than IIr~ @0| Z3| e which dominates IIA(T) IIe by (6). On general grounds, then, the adjoint A* is also contractive (with respect to I1" lie*)-

On the Clarkson-McCarthy Inequalities

11

Now we claim that A*((A + B ) G ( A + B ) O ( A - B ) O ( A - B)) = 2A| To see this we must check that for all T~ J o

2BQ0.

tr T(2A O 0 0 2 B O 0 ) = trA(T)((A + B ) O ( A + B ) O ( A - B ) O ( A - B)), that is

tr(2TIAOOO2T3BO0) r, W T 3

T,A-T3

Bearing in mind that tr(| = ~ trXk, verification of this is routine. Since A* is contractive, the inequality (7) follows. We have proved all the theorems stated in Sect. 2. We recall that the inequalities 2([[a [1~,+I[BI[~,)q/'< qlA+ Bl[q+ LqA-B[t%

(9)

(for 2=
(10)

(for 1 < p < 2 ) complement (1) and (2) to form the complete set of "ClarksonMcCarthy inequalities." We remark that, while (1) and (2) are commonly proved separately, they follow from (9) and (10) simply by the convexity properties of the power functions. Thus (1) is a consequence of (9) and the convexity of t ~ t p/q. It would therefore be doubly worthwhile to find a more direct proof (perhaps along the lines of our treatment of (1) and (2)) for the inequalities (9) and (10).

4. O n an Inequality o f Phillips

In [8] Phillips proved the following theorem, which is related to material in the preceding sections. Theorem 4 (J. Phillips). Let A >_B>O and t > 1. Then IIA'/'-B1/'II',<= IIA--BII1 9

(11)

When t = 2, this is a special case of the Powers-Stormer inequality [9] which is valid for any two positive operators A and B. Phillips gave an intricate proof of(11) and noted that a simpler proof would be possible if one had the inequality tr(At+ B')=__1. But this is a fact proved by McCarthy in [7]. Indeed, the inequalities (3) are equivalent to the inequalities 21 -P tr(A + B)P~ trAv + trB p < tr(A + B) p for all positive operators A, B and for all p > 1.

(12)

12

R. Bhatia and J. A. R. Holbrook

Let us n o w i n d i c a t e a s h o r t p r o o f of (11) f o l l o w i n g Phillips. T h e m a p A ~ A r is o p e r a t o r m o n o t o n e for 0 < r < 1 o n the class of positive o p e r a t o r s (see D o n o g h u e [4]). So, the c o n d i t i o n A _> B > 0 implies A 1/t > B1/t > 0 for all t > 1. U s i n g (l 2) write tr A = tr (A l/t _ B~/t + B 1/t)t ~ tr (A 1/' - B ~/~)'+ tr B. I.e., tr(A -- B) > tr (A l / t - B~/t) t, w h i c h is the same as the i n e q u a l i t y (11).

References 1. Ando, T.: Majorisation, doubly stochastic matrices and comparison of eigenvalues. Lecture Notes, Sapporo, 1982. Linear Algebra Appl. (to appear) 2. Bhatia, R.: Some inequalities for norm ideals. Commun. Math. Phys. 111, 33 39 (1987) 3. Bhatia, R., Holbrook, J.A.R.: Unitary invariance and spectral variation. Linear Algebra Appl. 95, 43--68 (1987) 4. Donoghue, W.: Monotone matrix functions and analytic continuation. Berlin, Heidelberg, New York: Springer 1974 5. Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear nonselfadjoint operators. Am. Math. Soc. Translations of Mathematical Monographs, Vol. 18 (1969) 6. Marshall, A.W., Olkin, 1.: Inequalities: Theory of majorisation and its applications. New York: Academic Press !979 7. McCarthy, C.A.: cp. Isr. J. Math. 5, 249 271 (1967) 8. Phillips, J.: Generalised Powers-Stormer inequalities. Talk at the Canadian Operator Theory Conference, Victoria, July 1986 (unpublished) 9. Powers, R.T., Stormer, E.: Free states of the canonical anticommutation relations. Commun. Math. Phys. 16, I 33 (1970) 10. Schatten, R.: Norm ideals of completely continuous operators (Sec. Ed.). Berlin, Heidelberg, New York: Springer 1970 11. Simon, B.: Trace ideals and their applications. Cambridge University Press, 1979 12. Thompson, R.C.: Convex and concave functions of singular values of matrix sums. Pac. J. Math. 62, 285 290 (1976)

Received February 24, 1987