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∗∗ ∈ X ∗∗ and y ∗ ∈ Y ∗, let x∗∗ ⊗ y ∗ be the functional deﬁned on ∗∗ ∗ ∗∗ ∗ ∗ ∗ ∗ L(X , Y ) by (x ⊗ y )(T ) = x (T (y )). For x0 ∈ X and y0 ∈ 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 ∈ L(X , Y ) ∗∗ ∗ ∗∗ ∗∗ ∗ ∗ kT k = sup{|(x ⊗ y )(T )| : x ∈ ∂e X1 , y ∈ ∂e Y1 }.

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 ∈ ∂e Y1 , y ∗(T (x)) = T ∗(y ∗)(x) = kT (x)k, so that kT (x)k ≤ kT ∗(y ∗)k. Similarly ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ ∗ let x0 ∈ ∂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 = CO{x ⊗ y : x ∈ ∂e X1 , y ∈ ∂e Y1 }.

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 = {x ⊗ y : x ∈ ∂e X1 , y ∈ ∂e Y1 }.

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 = CO{x ⊗ y : x ∈ ∂e X1 , y ∈ ∂e Y1 }. ∗ Let Λ ∈ ∂e K(X , Y )1. By Milman’s converse of the Krein-Milman’s ∗∗ ∗ ∗∗ ∗∗ ∗ ∗ theorem, Λ ∈ {x ⊗ y : x ∈ ∂e X1 , y ∈ ∂e Y1 }, where the closure is ∗ ∗∗ ∗∗ taken in the weak -topology. Thus there exists nets {xα } ⊂ ∂e X1 , ∗ ∗ ∗∗ ∗ ∗ {yα} ⊂ ∂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 ∈ X1 and y0 ∈ Y1 .

Let T ∈ 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 ∈ ∂e X1 and y0 ∈ ∂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 ∈ ∂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 ∈ X1 and y0 ∈ Y1 .

TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 9 / 19 ∗ ∗ ∗ ∗ ∗ Now for a T ∈ 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 ∈ L(X , Y ). We know from an application of the ∗ Krein-Milman and Hahn-Banach theorems, for some Λ ∈ ∂e L(X , Y )1. ∗ ∗ ∗ Λ(T ) = kT k. Now suppose Λ = x0 ⊗ y0 for some x0 ∈ ∂e X1, y0 ∈ ∂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 ∈ {L(X , Y ): T or T ∗ attains norm} 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 {xn }n≥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 {tn}n≥1 ⊂ K and pair-wise disjoint open sets Un, fn ∈ C(K), 0 ≤ fn ≤ 1, fn(tn) = 1 and fn(K − Un) = 0. P Define φ : c0 → C(K) by φ({αn}n≥1) = αnfn. It is easy to see that φ is a linear isometry. ∗ Now T : X → C(K)defined by T (x) = φ({xn (x)}n≥1) is an operator that is not compact. ∗ More generally for any {βn}n≥1, the mapping S(x) = φ({βnxx (x)}n≥1 gives an isometric copy of `∞ 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 ∈ W ∗C(K, X ) ∗ there is a sequence {fn}n≥1 ⊂ C(K, X ) such that fn(k) → f (k) for all

k ∈ 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 ∗). We recall from the theory of Vector Measures (Diestel and Uhl, AMS Surveys), a theorem of Singer, that identifies C(K, X ∗)∗ as the space M(K, X ∗∗) of X ∗∗-valued, regular Borel vector measures of finite total variation, equipped with the total variation norm.

∗∗ ∗ R Let Λ ∈ M(K, X ), let F = Λ|C(K, X ), deﬁne P(Λ)(f ) = limn K fndF , ∗ where fn ∈ C(K, X ) converge point-wise to f and the limit of the integrals exists by DCT.

TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 16 / 19 It is easy to see that P is a well-deﬁned linear contraction. Clearly if Λ ∈ C(K, X ∗)⊥, P(Λ) = 0. On the other hand ifP(Λ) = 0, then for ∗ R ∗ f ∈ C(K, X ), P(Λ)(f ) = 0 = k fdF , thus F = Λ|C(K, X ) = 0. Hence ker(P) = C(K, X ∗)⊥.

To see that P is a projection, note that P(Λ)|C(K, X ∗) = F .

TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 17 / 19 ∗∗ ∗∗ We next note that for x ∈ ∂e X1 , k ∈ K,

∗∗ ∗∗ ∗∗ ∗∗ P(x ⊗ δ(k))(f ) = lim x (fn(k)) = x (f (k)) = (x ⊗ δ(k))(f ). n

Thus P(x∗∗ ⊗ y ∗) = x∗∗ ⊗ y ∗. Since functionals of this type determine the norm of an operator, it is easy to see that under the canonical embedding,

C(K, X ∗) ⊂ W ∗C(K, X ∗) ⊂ C(K, X ∗)∗∗.

We already saw that for an infinite set K and an infinite dimensional space X , the first inclusion is proper.

TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 18 / 19 We next show that the second embedding is also not onto. Since K is infinite, let B ⊂ K be a Borel set such that χB is not a continuous ∗ ∗ ∗ function. Let x0 ∈ X1 and x0 ∈ X1 be such that x0 (x0) = 1. Recall Singer’s theorem we have the identification of C(K, X ∗)∗ as the space of regular vector measures M(K, X ∗∗). Now consider the functional ∗ ∗∗ ∗ ∗ χB ⊗ x0 ∈ C(K, X ) defined by (χB ⊗ x0 )(F ) = F (B)(x0 ) for ∗∗ ∗ ∗ ∗ F ∈ M(K, X ). Suppose f ∈ W C(K, X ) is such that f = χB ⊗ x0 . It is easy to see that χB = x0 ◦ f . This is a contradiction.

Now it is natural to ask, if there can be a surjective isometry between K(X , C(K))∗∗ and L(X , C(K)). Partial negative answers can be given using the theory of centralizers and results from the work of M. Cambern and P. Greim. It seems harder to decide if K(X , C(K))∗∗ can be isometric to L(Z, C(∆)) for some Banach space Z and extremally disconnected compact Hausdorﬀ space ∆?

TSSRK Rao (ISI, Bangalore) A quick introduction to the geometry of spaces of operators April 25, 2018 19 / 19