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An Introduction to the Geometry of Spaces of Operators on Banach A quick introduction to the geometry of spaces of operators T. S. S. R. K. Rao Theoretical Statistics and Mathematics Unit Indian Statistical Institute R. V. College P.O. Bangalore 560059 India,E-mail : [email protected], [email protected] April 25, 2018 TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 1 / 19 Abstract For Banach spaces X ; Y we give a brief account of the geometry of the dual unit ball of the space of bounded linear operators L(X ; Y ) and attempt the near impossible task of understanding the bidual L(X ; Y )∗∗ using bare hands approach. TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 2 / 19 An important problem in the theory of operators on a Banach space is to understand the relative position of the space of compact operators K(X ; Y ) in the space of bounded operators. Throughout these lectures we assume that X ; Y are infinite dimensional Banach spaces (most often also non-reflexive) and there is a non-compact operator in L(X ; Y ). For a Banach space X let X1 denote the closed unit ball, @e X1 denote the set of extreme points. We always consider a non-reflexive Banach space as canonically embedded in its bidual X ∗∗. TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 3 / 19 For x∗∗ 2 X ∗∗ and y ∗ 2 Y ∗, let x∗∗ ⊗ y ∗ be the functional defined on ∗∗ ∗ ∗∗ ∗ ∗ ∗ ∗ L(X ; Y ) by (x ⊗ y )(T ) = x (T (y )). For x0 2 X and y0 2 Y by ∗∗ ∗ ∗ x0 ⊗ y0 we denote the operator (x0 ⊗ y0)(x) = x0 (x)y0. It is easy to see ∗ ∗ ∗ that kx0 ⊗ y0k = kx0 kky0k. Thus X and Y are isometric to subspaces of K(X ; Y ). The following key Lemma gives a formula for kSk. Lemma For T 2 L(X ; Y ) ∗∗ ∗ ∗∗ ∗∗ ∗ ∗ kT k = supfj(x ⊗ y )(T )j : x 2 @e X1 ; y 2 @e Y1 g. TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 4 / 19 Proof. For > 0, let kxk = 1 and kT (x)k > kT k − . By an application of the ∗ ∗ Krein-Milman's theorem, let y 2 @e Y1 , y ∗(T (x)) = T ∗(y ∗)(x) = kT (x)k, so that kT (x)k ≤ kT ∗(y ∗)k. Similarly ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ ∗ let x0 2 @e X1 be such that x0 (T (y )) = kT (y )k so that ∗∗ ∗ (x0 ⊗ y0 )(T ) > kT k − . TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 5 / 19 Now an application of separation theorem gives ∗ ∗∗ ∗ ∗∗ ∗∗ ∗ ∗ L(X ; Y )1 = COfx ⊗ y : x 2 @e X1 ; y 2 @e Y1 g. The next result is attributed to Ruess and Stegall and we only indicate the proof of the easy part of the theorem. W. Ruess and C. Stegall : Extreme points in duals of operator spaces, Math. Ann. 261 (1982) 535-546. Theorem Let X ; Y be Banach spaces. ∗ ∗∗ ∗ ∗∗ ∗∗ ∗ ∗ @e K(X ; Y )1 = fx ⊗ y : x 2 @e X1 ; y 2 @e Y1 g. TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 6 / 19 proof Note that since the functionals x∗∗ ⊗ y ∗ are defined on both K(X ; Y ) as well as L(X ; Y ), the above arguments work verbatim to give ∗ ∗∗ ∗ ∗∗ ∗∗ ∗ ∗ K(X ; Y )1 = COfx ⊗ y : x 2 @e X1 ; y 2 @e Y1 g. ∗ Let Λ 2 @e K(X ; Y )1. By Milman's converse of the Krein-Milman's ∗∗ ∗ ∗∗ ∗∗ ∗ ∗ theorem, Λ 2 fx ⊗ y : x 2 @e X1 ; y 2 @e Y1 g, where the closure is ∗ ∗∗ ∗∗ taken in the weak -topology. Thus there exists nets fxα g ⊂ @e X1 , ∗ ∗ ∗∗ ∗ ∗ fyαg ⊂ @e Y1 such that xα ⊗ yα ! Λ in the weak -topology. TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 7 / 19 Now using the weak∗-compactness of the dual unit balls, we may assume ∗∗ ∗∗ ∗ ∗ ∗ w. l. o. g that xα ! x0 and yα ! y0 in the weak -topology, for some ∗∗ ∗∗ ∗ ∗ x0 2 X1 and y0 2 Y1 . Let T 2 K(X ; Y ), since T ∗ is a compact operator, we get, ∗ ∗ ∗ ∗ T (yα) ! T (y0 ) in the norm. Now it is easy to see that ∗∗ ∗ ∗ ∗∗ ∗ ∗ xα (T (yα)) ! x0 (T (y )). ∗∗ ∗ ∗∗ ∗ ∗∗ ∗ Thus (xα ⊗ yα)(T ) ! (x0 ⊗ y0 )(T ). Hence Λ = x0 ⊗ y0 . Since Λ is an ∗∗ ∗∗ ∗ ∗ extreme point, it is easy to see that x0 2 @e X1 and y0 2 @e Y1 . This completes the proof of one part of the theorem. TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 8 / 19 Temptation Why doesn't this proof work in L(X ; Y )? ∗ Let Λ 2 @e L(X ; Y )1. We can still apply Milman's converse. As before get ∗∗ ∗ ∗ nets, such that xα ⊗ yα ! Λ in the weak -topology. ∗∗ ∗∗ ∗ ∗ We may as before assume w. l. o. g that xα ! x0 and yα ! y0 in the ∗ ∗∗ ∗∗ ∗ ∗ weak -topology, for some x0 2 X1 and y0 2 Y1 . TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 9 / 19 ∗ ∗ ∗ ∗ ∗ Now for a T 2 L(X ; Y ), T (yα) ! T (y0 ) only in the weak -topology. ∗∗ ∗ ∗ So we have no control on xα (T (yα)). ∗∗ ∗ We don't know if x0 and/or y0 are non-zero? We will see examples where such a Λ actually vanishes on all compact operators. So such a Λ can't be like x∗∗ ⊗ y ∗ If you are restless, try this for X = Y = `2. TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 10 / 19 Reason Let X be reflexive. Let T 2 L(X ; Y ). We know from an application of the ∗ Krein-Milman and Hahn-Banach theorems, for some Λ 2 @e L(X ; Y )1. ∗ ∗ ∗ Λ(T ) = kT k. Now suppose Λ = x0 ⊗ y0 for some x0 2 @e X1, y0 2 @e Y1 . ∗ ∗ ∗ Thus T (y0 )(x) = y0 (T (x0)) = kT k. Therefore kT k = kT (x0)k and ∗ ∗ ∗ ∗ kT k = kT (y0 )k, i.e., T and T attain norm. Study of richness of the set of operators that attain the norm is an ever popular area in Operator theory. TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 11 / 19 Arguments given here lead to a proof of the easy fact: For a compact operator T , T ∗ always attains its norm. See : Martin, Miguel Norm-attaining compact operators. J. Funct. Anal. 267 (2014), no. 5, 15851592. for recent update. We end the general discussion by recalling an old result: V. Zizler : On some extremal problems in Banach spaces, Math. Scand. 32 (1973) 214-224 (1974). T 2 fL(X ; Y ): T or T ∗ attains normg is dense in L(X ; Y ). TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 12 / 19 Y = C(K) Let K be a compact Hausdorff space and let C(K) denotes the space of real-valued continuous functions on K, equipped with the supremum norm. Since any Banach space Y isometrically embeds into a C(K) space, we now consider L(X ; C(K)) for an infinite dimensional Banach space X and infinite compact set K. Since X is an infinite dimensional space, by the Josefson{Nissenzweig ∗ theorem (J. Diestel' Springer GTM 92) there exists a sequence fxn gn≥1 of ∗ ∗ unit vectors such that xn ! 0in the weak -topology. TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 13 / 19 Since K is an infinite set, choose a sequence ftngn≥1 ⊂ K and pair-wise disjoint open sets Un, fn 2 C(K), 0 ≤ fn ≤ 1, fn(tn) = 1 and fn(K − Un) = 0. P Define φ : c0 ! C(K) by φ(fαngn≥1) = αnfn. It is easy to see that φ is a linear isometry. ∗ Now T : X ! C(K)defined by T (x) = φ(fxn (x)gn≥1) is an operator that is not compact. ∗ More generally for any fβngn≥1, the mapping S(x) = φ(fβnxx (x)gn≥1 gives an isometric copy of `1 in L(X ; C(K). TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 14 / 19 Function space formulation ∗ Let δ : K ! C(K)1 be the evaluation (or Dirac measure) map that is a homeomorphism when the range is equipped with the weak∗-topology. Let W ∗C(K; X ∗) denote the space of functions continuous w. r. t the weak∗-topology, equipped with the supremum norm. Define Φ: L(X ; C(K)) ! W ∗C(K; X ∗) by Φ(T ) = T ∗ ◦ δ. It is fairly routine to verify that Φ is a surjective isometry that maps K(X ; C(K)) onto C(K; X ∗) (space of norm continuous functions). From now on we work in this set up. Let us make the following topological assumption. For f 2 W ∗C(K; X ) ∗ there is a sequence ffngn≥1 ⊂ C(K; X ) such that fn(k) ! f (k) for all k 2 K, kfnk ≤ kf k for all n. TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 15 / 19 We are now ready to define a projection on W ∗C(K; X ∗).
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