Which multiplication operators are surjective isometries?
aa r X i v : . [ m a t h . F A ] J a n Which multiplication operators are surjectiveisometries?
Eugene Bilokopytov ∗ January 29, 2019
Abstract
Let F be a Banach space of continuous functions over a connected locallycompact space X . We present several sufficient conditions on F guaranteeingthat the only multiplication operators on F that are surjective isometries arescalar multiples of the identity. The conditions are given via the properties ofthe inclusion operator from F into C ( X ), as well as in terms of geometry of F .An important tool in our investigation is the notion of Birkhoff Orthogonality. Keywords:
Function Spaces, Multiplication Operators, Surjective Isome-tries, Birkhoff Orthogonality, Nearly Strictly Convex Spaces;MSC2010 46B20, 46E15, 47B38
Normed spaces of functions are ubiquitous in mathematics, especially in analysis.These spaces can be of a various nature and exhibit different types of behavior,and in this work we discuss some questions related to these spaces from a general,axiomatic viewpoint. The class of linear operators that capture the very nature ofof the spaces of functions is the class of weighted composition operators (WCO).Indeed, the operations of multiplication and composition can be performed on anycollection of functions, while there are several Banach-Stone-type theorems whichshow that the WCO’s are the only operators that preserve various kinds of structure(see e.g. [13] and [15] for more details).In this article we continue our investigation (see [7]) of the general frameworkwhich allows to consider any Banach space that consists of continuous (scalar-valued)functions, such that the point evaluations are continuous linear functionals, and ofWCO’s on these spaces.First, let us define precisely what we mean by a normed space of continuousfunctions. Let X be a topological space (a phase space ) and let C ( X ) denote thespace of all continuous complex-valued functions over X endowed with the compact-open topology. A normed space of continuous functions (NSCF) over X is a linearsubspace F ⊂ C ( X ) equipped with a norm that induces a topology, which is stronger ∗ Email address [email protected], [email protected]. J F : F → C ( X ) iscontinuous, or equivalently the unit ball B F is bounded in C ( X ). If F is a linearsubspace of C ( X ), then the point evaluation at x ∈ X on F is the linear functional x F : F → C , defined by x F ( f ) = f ( x ). If F is a NSCF, then all point evaluationsare bounded on F . Conversely, if F ⊂ C ( X ) is equipped with a complete norm suchthat x F ∈ F ∗ , for every x ∈ X , then F is a NSCF. We will call a NSCF F over X (weakly) compactly embedded if J F is a (weakly) compact operator, or equivalently,if B F is (weakly) relatively compact in C ( X ).Let X and Y be topological spaces, and let Φ : Y → X and ω : Y → C (notnecessarily continuous). A weighted composition operator (WCO) with compositionsymbol Φ and multiplicative symbol ω is a linear map W Φ ,ω from the space of allcomplex-valued functions on X into the analogous space over Y defined by[ W Φ ,ω f ] ( y ) = ω ( y ) f (Φ ( y )) , for y ∈ Y . Let F ⊂ C ( X ), E ⊂ C ( Y ) be linear subspaces. If W Φ ,ω F ⊂ E , thenwe say that W Φ ,ω is a weighted composition operator from F into E (we use thesame notation W Φ ,ω for what is in fact W Φ ,ω | F ). In particular, if X = Y , we willdenote M ω = W Id X ,ω , and call it the multiplication operator (MO) with symbol(or weight ) ω . If in this case F = E , then we will call ω a multiplier of F . If F and E are both complete NSCF’s, then any WCO between these spaces is automaticallycontinuous due to Closed Graph theorem. However, in concrete cases it can be verydifficult to determine all WCO’s between a given pair of NSCF’s. In particular, itis difficult to determine all multipliers of a NSCF (see e.g. [21] and [26], where themultiplier algebras of some specific families of NSCF’s are described).WCO’s may be viewed as morphisms in the category of NSCF’s. In the lightof this fact it is important to be able to characterize WCO’s with some specificproperties. In this article we focus on one such property – being a unitary , i.e. asurjective isometry, or an isometric isomorphism. More specifically, we consider thefollowing rigidity property of a NSCF F over X : if E is a NSCF over Y , ω, υ : Y → C and Φ : Y → X are such that both W Φ ,ω and W Φ ,υ are unitaries from F into E , thenthere is λ ∈ C , | λ | = 1 with υ = λω . In particular, we are looking for conditions on F such that the only unitary MO’s on F are the scalar multiples of the identity.Some related problems were studied (see e.g. [1], [2], [3], [8], [18], [19], [20], [27],[28] and [29]). Note that in these articles the class of operators under considerationis wider (e.g. unitary WCO’s, or isometric MO’s, as opposed to unitary MO’s), butthese operators are considered on the narrower classes of NSCF’s.Let us describe the contents of the article. In Section 2 we gather some ele-mentary properties of NSCF’s and WCO’s. In particular, we characterize weaklycompactly embedded NSCF’s (Theorem 2.3) and prove that a WCO between com-plete NSCF’s with a surjective composition symbol is a linear homeomorphism ifand only if its adjoint is bounded from below (part (iii) of Corollary 2.13). Section3 is dedicated to the main problem of the article, and in particular it contains themain results (Theorem 3.9 and Proposition 3.14), which give sufficient conditions If X is a set, then by Id X we denote the identity map on X . or a NSCF to have the rigidity properties described above. In Section 4 we con-sider an interpretation of Theorem 3.9 for abstract normed spaces, as opposed toNSCF’s. Also, we study some properties of Birkhoff (-James) orthogonality whichis an important tool in our investigation. Finally, we consider the class of nearlystrictly convex normed spaces that includes strictly convex and finitely dimensionalnormed spaces, and arises naturally when studying NSCF’s in the context of Birkhofforthogonality. Some notations and conventions.
Let D (or D ) be the open (or closed) unitdisk on the plane C , and let T = ∂ D be the unit circle. For a linear space E let E ′ be the algebraic dual of E , i.e. the linear space of all linear functionals on E . In this section we discus some basic properties of NSCF’s and WCO’s. Let us startwith NSCF’s. We will often need to put certain restrictions on the phase spaces ofNSCF’s. A Hausdorff topological space X is called compactly generated , or a k-space whenever each set which has closed intersections with all compact subsets of X isclosed itself. It is easy to see that all first countable (including metrizable) and alllocally compact Hausdorff spaces are compactly generated. Moreover, Arzela-Ascolitheorem describes the compact subsets of C ( X ) in the event when X is compactlygenerated, which further justifies the importance of this class of topological spaces.Details concerning the mentioned facts and some additional information about thecompactly generated spaces can be found in [10, 3.3].Let us characterize (weakly) compactly embedded NSCF’s using the followingvariation of a classic result (see [6], [9, VI.7, Theorem 1], [16, 3.7, Theorem 5], [25]). Theorem 2.1.
Let F be a NSCF over a Hausdorff space X . Then κ F is a weak*continuous map from X into F ∗ . Moreover, the following equivalences hold:(i) F is weakly compactly embedded if and only if κ F is weakly continuous.(ii) If κ F is norm-continuous, then F is compactly embedded. The converses holdswhenever X is compactly generated. More generally, every linear map T from a linear space F into C ( X ) generatesa weak* continuous map κ T : X → F ′ defined by h f, κ T ( x ) i = [ T f ] ( x ), for x ∈ X and f ∈ F . In this case κ T ( A ) ⊥ = κ T (cid:0) A (cid:1) ⊥ , for any A ⊂ X , and Ker T = κ T ( X ) ⊥ . Remark . Clearly, every compactly embedded NSCF’s is weakly compactly em-bedded. On the other hand, it follows from the theorem above that any reflexiveNSCF is also weakly compactly embedded.If X is a domain in C n , i.e. an open connected set, and F is a NSCF over X that consists of holomorphic functions, then F is compactly embedded. Indeed, byMontel’s theorem (see [24, Theorem 1.4.31]), B F is relatively compact in C ( X ), sinceit is a bounded set that consists of holomorphic functions.For a NSCF F over X let B F C ( X ) be the closure of B F in C ( X ). Since B F C ( X ) is bounded, closed, convex and balanced, we can generate a NSCF with the closednit ball B F C ( X ) . Namely, define b F = n αf (cid:12)(cid:12)(cid:12) α > , f ∈ B F C ( X ) o , which is a linearsubspace of C ( X ), and endow it with the norm being the Minkowski functionalof B F C ( X ) . Since B b F = B F C ( X ) is bounded in C ( X ), it follows that b F is a NSCFover X . It is clear that F is (weakly) compactly embedded if and only if b F is(weakly) compactly embedded. It turns out, that the fact that F is weakly compactlyembedded can be further characterized in terms of b F and B F C ( X ) . Theorem 2.3.
Let F be a NSCF over a Hausdorff space X . Then the following areequivalent:(i) F is weakly compactly embedded;(ii) B F C ( X ) is compact with respect to the pointwise topology on C ( X ) ;(iii) b F = (span κ F ( X )) ∗ (as normed spaces) via the bilinear form induced by h x F , f i = f ( x ) .Proof. (iii) ⇒ (ii): If (iii) holds, then the pointwise topology on b F coincides with theweak* topology. Hence, the unit ball B F C ( X ) is pointwise compact due to Banach-Alaoglu theorem.(ii) ⇔ (i): From the definition of a NSCF, B F is bounded in C ( X ). Hence, thisset is weakly relatively compact if and only if it is relatively compact with respectto the pointwise topology on C ( X ) (see [14, 4.3, Corollary 2]).(i) ⇒ (iii): If F is weakly compactly embedded then J F is weakly compact, andso J ∗∗ F maps F ∗∗ into C ( X ) with J ∗∗ F B F ∗∗ = B F C ( X ) = B b F (the proof of [9, VI.4,Theorem 2] carries over to the case when the target space is locally convex, see also[16, 2.18, Theorem 13]). Consequently, J ∗∗ F B F ∗∗ = B b F . Indeed, if f ∈ B b F , there is g ∈ B F ∗∗ such that J ∗∗ F g = f k f k b F . Then J ∗∗ F (cid:0) k f k b F g (cid:1) = f , and since k g k ≤ k f k b F < f ∈ J ∗∗ F B F ∗∗ . On the other hand, as k J ∗∗ F k L ( F ∗∗ , b F ) ≤
1, itfollows that J ∗∗ F g ∈ B b F , for any g ∈ B F ∗∗ .Hence, J ∗∗ F is a quotient map from F ∗∗ onto b F (see the proof of [17, Lemma2.2.4]), and so b F ≃ F ∗∗ / Ker J ∗∗ F . For g ∈ F ∗∗ we have that g ∈ Ker J ∗∗ F if andonly if [ J ∗∗ F g ] ( x ) = 0, for every x ∈ X . By definition, [ J ∗∗ F g ] ( x ) = h g, x F i , and soKer J ∗∗ F = κ F ( X ) ⊥ in F ∗∗ . Finally, since F ∗∗ /κ F ( X ) ⊥ is isometrically isomorphicto (span κ F ( X )) ∗ (see the proof of [11, Proposition 2.6]), the result follows. Corollary 2.4.
Let F be a NSCF over a Hausdorff space X . Then F = (span κ F ( X )) ∗ (as normed spaces) if and only if F is weakly compactly embed-ded and B F is closed in C ( X ) . Let us consider some examples of NSCF’s.
Example . Let C ∞ ( X ) be the space of all bounded continuous functions on X ,with the supremum norm k f k = sup x ∈ X | f ( x ) | . It is easy to see that C ∞ ( X ) is acomplete NSCF, but if X is not a discrete topological space, then C ∞ ( X ) is NOTweakly compactly embedded. Indeed, its closed unit ball C (cid:0) X, D (cid:1) is not a pointwisecompact set since any f : X → D can be approximated by elements of C (cid:0) X, D (cid:1) inthe pointwise topology. xample . Let (
X, d ) be a metric space and let z ∈ X . For f : X → C definedil f = sup n | f ( x ) − f ( y ) | d ( x,y ) | x, y ∈ X, x = y o . This functional generates a NSCF Lip ( X, d ) = { f : X → C | dil f < + ∞ } with the norm k f k = dil f + | f ( z ) | . Onecan show that F = Lip ( X, d ) is a complete NSCF with k x F k = max { , d ( x, z ) } and k x F − y F k = d ( x, y ), for every x, y ∈ X (the proof is a slight modification ofthe proof from [5]). Hence, Lip ( X, d ) is compactly embedded due to part (ii) ofTheorem 2.1. Moreover, it not difficult to show that B F is closed in C ( X ), and so F = (span κ F ( X )) ∗ , due to Corollary 2.4.Let us now consider basic properties of WCO’s and in particular MO’s. We startwith a well-known fact (see e.g. [7, Proposition 2.4 and Corollary 2.5]). Proposition 2.7.
Let X and Y be topological spaces. Let F ⊂ C ( X ) and E ⊂ C ( Y ) be linear subspaces, an let T be a linear map from F into E . Then T = W Φ ,ω , for Φ : Y → X and ω : Y → C if and only if T ′ κ E ( y ) = ω ( y ) κ F (Φ ( y )) , for every y ∈ Y . In other words, T is a WCO if and only if T ′ κ E ( Y ) ⊂ C κ F ( X ) . In particular, if X = Y and F = E , then T is a MO if and only if x F is aneigenvector of T (or else x F = 0 F ′ ), for every x ∈ X . Then the multiplier is thecorrespondence between x and the eigenvalue of T for x F . Also, it follows thatKer W Φ ,ω = (cid:0) W ′ Φ ,ω κ E ( Y ) (cid:1) ⊥ = κ F (cid:0) Φ (cid:0) Y \ ω − (0) (cid:1)(cid:1) ⊥ = κ F (cid:16) Φ ( Y \ ω − (0)) (cid:17) ⊥ . Note that in general we cannot reconstruct the symbols of a WCO from its dataas a linear operator between certain NSCF’s, in the sense that the equality of WCO’sdoes not imply the equality of their symbols.
Example . Let F and E be NSCF’s over topological spaces X and Y respectively. • If x ∈ X is such that x F = 0 F ∗ , i.e. f ( x ) = 0, for every f ∈ F , then M ω on F does not depend on ω ( x ), in the sense that if ω, υ : X → C coincide outsideof x , then M ω = M υ . • If ω ( y ) = 0, for y ∈ Y , then W Φ ,ω does not depend on Φ ( y ), for Φ : Y → X . • More generally, we can construct a WCO with nontrivial symbols which isequal to the identity on F if there are two distinct points in X such that thepoint evaluations on F at these points are linearly dependent.Since we are interested in investigating properties of the symbols of WCO’s basedon their operator properties, we need to be able to reconstruct the symbols. Hence,we have to introduce the following concepts. We will call a linear subspace F of C ( X ) 1- independent if 0 F ′ κ F ( X ), i.e. for every x ∈ X there is f ∈ F such that f ( x ) = 0. We will say that F is 2- independent if x F and y F are linearly independent,for every distinct x, y ∈ X . It is easy to see that this condition is equivalent to theexistence of f, g ∈ F such that f ( x ) = 1, f ( y ) = 0, g ( x ) = 0 and g ( y ) = 1.Note that if F is 2-independent, it is 1- independent and separates points of X , if F contains nonzero constant functions, it is 1-independent, and if F contains nonzeroconstant functions and separates points, it is 2-independent. However, the conversesto these statements do not hold.t is easy to see that MO’s from a 1-independent NSCF determine their sym-bols, and WCO’s from a 2-independent NSCF also determine their symbols (see[7, Proposition 2.8]). Moreover, some properties of the symbols of WCO can indeedbe recovered (see [7, Corollary 3.3 and Proposition 4.3]). Proposition 2.9.
Let F be a NSCF over a topological space X . Then:(i) If F is -independent, then its multipliers are continuous.(ii) If X is a domain in C n , and F consists of holomorphic functions, then its multi-pliers can be chosen to be holomorphic, in the sense that if T : F → F is a continuousMO, then there is a holomorphic ω : X → C such that T = M ω . The following examples demonstrates that we cannot relax the requirement of1-independence in part (i).
Example . Let D ⊂ C be endowed with the usual metric. Let F = (cid:8) f ∈ Lip (cid:0) D (cid:1) | f (0) = 0 (cid:9) with the norm k f k = dil f , f ∈ F . This is a com-plete compactly embedded NSCF, and the set (cid:8) x ∈ D | x F = 0 F ∗ (cid:9) is the singleton { } . Define ω : D → T by ω ( z ) = z | z | , when z = 0 and ω (0) = 1. Clearly, ω hasa non-removable discontinuity at 0. On the other hand, let us show that M ω is abounded invertible operator on F .Let f ∈ F and denote g = M ω f . First, g (0) = 0 and | g ( z ) − g (0) | = | f ( z ) | = | f ( z ) − | ≤ | z − | dil f, for every z ∈ D \ { } . Furthermore, for distinct z, y ∈ D \ { } with | z | ≥ | y | we get | g ( z ) − g ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12) z | z | f ( z ) − y | y | f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) z | z | (cid:12)(cid:12)(cid:12)(cid:12) | f ( z ) − f ( y ) | + (cid:12)(cid:12)(cid:12)(cid:12) z | z | − y | y | (cid:12)(cid:12)(cid:12)(cid:12) | f ( y ) | . We have (cid:12)(cid:12)(cid:12) z | z | (cid:12)(cid:12)(cid:12) | f ( z ) − f ( y ) | = | f ( z ) − f ( y ) | ≤ | z − y | dil f . At the same time, (cid:12)(cid:12)(cid:12) z | z | − y | y | (cid:12)(cid:12)(cid:12) | f ( y ) | ≤ (cid:12)(cid:12)(cid:12) z | z | − y | y | (cid:12)(cid:12)(cid:12) | y | dil f = (cid:12)(cid:12)(cid:12) | y | z | z | − y (cid:12)(cid:12)(cid:12) dil f , and using | z | ≥ | y | it notdifficult to prove that (cid:12)(cid:12)(cid:12) | y | z | z | − y (cid:12)(cid:12)(cid:12) ≤ | z − y | . Hence, | g ( z ) − g ( y ) | ≤ | z − y | dil f ,and as y and z were chosen arbitrarily we conclude that dil g ≤ f , and so k M ω f k = k g k ≤ k f k . Since f was chosen arbitrarily, we get k M ω k ≤
2. Asdil f = dil f , for any f ∈ F , it follows that k M ω k = k M ω k ≤
2, and since w = ω weobtain k M − ω k = k M ω k = k M ω k ≤ Example . Let D and ω be as in the previous example. For N = n ∈ N ∪{ } con-sider e n : D → C defined by e n ( z ) = z ω ( z ) n n . Note that { e n } n ∈ N is linearly indepen-dent (allowing infinite series), and so there is a compactly embedded Hilbert NSCF E whose orthonormal basis is { e n } n ∈ N . Namely, E is the Reproducing Kernel Hilbertspace generated by the positive semi-definite kernel K ( z, w ) = P n ∈ N e n ( z ) e n ( w ) = zw | z || w |− zw (see e.g. [12]). Then M ω acts as a unilateral shift (and in particular is anisometry) on E , and so k M ω k = 2.Furtheremore, using { e n } n ∈ Z , where e n : D → C is defined by e n ( z ) = z ω ( z ) n | n | onecan construct a compactly embedded Hilbert NSCF for which M ω is an invertibleoperator (but not a scalar multiple of an isometry).et us now derive some properties of WCO’s from the properties of their symbols. Proposition 2.12.
Let X and Y be topological spaces and let F ⊂ C ( X ) be a linearsubspace. Let Φ , Ψ : Y → X be continuous and let ω, υ : Y → C be such that W Φ ,ω F ⊂ C ( Y ) and W Ψ ,υ F ⊂ C ( Y ) . Then:(i) If Φ has a dense image and ω vanishes on a nowhere dense set, then W Φ ,ω is aninjection (cf. [7, Proposition 2.6] ).(ii) Assume that there is a linear operator T : F → F such that W Ψ ,υ = W Φ ,ω T . If Φ is a surjection, ω vanishes on a nowhere dense set, and there is a continuous function η : Y → C , such that υ = ηω , then there are maps Θ : X → X and θ : X → C suchthat T = W Θ ,θ . If F is -independent, then Θ ◦ Φ = Ψ and θ ◦ Φ = η .Proof. Let Z = Y \ ω − (0), which is a dense subset of Y .(i): If Φ has a dense image, then Φ ( Z ) = Φ (cid:0) Z (cid:1) = Φ ( Y ) = X , and soKer W Φ ,ω = κ F (cid:16) Φ ( Z ) (cid:17) ⊥ = κ F ( X ) ⊥ = { } , since κ F ( X ) is separating on F .(ii): If W Ψ ,υ = W Φ ,ω T then T ′ W ′ Φ ,ω = W ′ Ψ ,υ , and so ω ( y ) T ′ Φ ( y ) F = υ ( y ) Ψ ( y ) F ,for every y ∈ Y . Hence, T ′ Φ ( y ) F = η ( y ) Ψ ( y ) F , for each y ∈ Z .Note that both T ′ ◦ κ F ◦ Φ and η · κ F ◦ Ψ are weak* continuous maps from Y into F ′ .Indeed, the adjoint operator is always continuous with respect to the weak* topology,while κ F ◦ Φ and κ F ◦ Ψ are compositions of continuous maps; finally, multiplyinga weak* continuous map with a continuous function is weak* continuous since theweak* topology is linear.Hence, T ′ ◦ κ F ◦ Φ and η · κ F ◦ Ψ are weak* continuous maps from Y into F ′ thatcoincide on a dense set Z , and so T ′ Φ ( y ) F = η ( y ) Ψ ( y ) F , for every y ∈ Y . As Φ isa surjection we get that T ′ κ F ( X ) ⊂ C κ F ( X ), and so by virtue of Proposition 2.7, T is a WCO, i.e. T = W Θ ,θ , for some Θ : X → X and θ : X → C . Since in thiscase W Ψ ,υ = W Φ ,ω W Θ ,θ = W Θ ◦ Φ ,ω · θ ◦ Φ , if F is 2-independent, then Θ ◦ Φ = Ψ and θ ◦ Φ = η . Corollary 2.13.
Let F be a NSCF over a topological space X , let E be a -independent NSCF over a topological space Y , and let Φ : Y → X and ω : Y → C be such that W Φ ,ω ∈ L ( F , E ) . Then:(i) If W ∗ Φ ,ω is an injection, then ω does not vanish.(ii) If Φ has a dense image and W ∗ Φ ,ω is an injection, then W Φ ,ω is an injection.(iii) If F and E are Banach spaces and Φ has a dense image, then W ∗ Φ ,ω is boundedfrom below (isometry) if and only if W Φ ,ω is an linear homeomorphism (unitary).Proof. (i),(ii): If W ∗ Φ ,ω is an injection, then H = W Φ ,ω F is dense in E . One canshow that a dense subspace of a 1-independent NSCF is 1-independent. Hence, if ω ( y ) = 0, then y H = 0, which leads to a contradiction. If in this case Φ has a denseimage, then W Φ ,ω is an injection (see [7, Proposition 2.6]).(iii): We only need to show sufficiency. Assume that W ∗ Φ ,ω is bounded from below.Then it follows from part (ii) that W Φ ,ω is an injection with a dense image. However,since W ∗ Φ ,ω is bounded from below it follows that the image of W Φ ,ω is closed (see [11,Exercise 2.49] with the solution therein). Hence, W Φ ,ω is a linear homeomorphism,nd so W ∗ Φ ,ω is also a linear homeomorphism (see the same reference). If in this case W ∗ Φ ,ω is an isometry, then it is a unitary, and so W Φ ,ω is also a unitary. In this section we investigate our main question. Namely, we look for conditions ona NSCF that would prevent it from admitting unitary MO’s other than the scalarmultiples of the identity. Let us first consider some examples of such conditions.
Example . Assume that X is a domain in C n and F = { } is a NSCF over X thatconsists of holomorphic functions on X . Let ω : X → C be such that M ω is unitaryon F . From part (ii) of Proposition 2.9 we may assume that ω is holomorphic on X . Since M ω is unitary, M ∗ ω is an isometry on F ∗ , and so from Proposition 2.7 itfollows that | ω ( x ) | = 1 for every x ∈ X such that x F = 0 F ∗ . Let f ∈ F \ { } .Then for every x f − (0) we have that x F = 0 F ∗ , and so | ω ( x ) | = 1. Hence, ω is holomorphic on X and such that | ω | ≡ X \ f − (0).From the Open Mapping theorem (see [24, Conclusion 1.2.12]) it follows that ω is aconstant function. Remark . If we dealt with real-valued functions, then ω would be real-valued.Hence, if F was a 1-independent “real-valued” NSCF over a connected space X , thenfrom part (i) of Proposition 2.9, ω would be a continuous function on a connectedspace with valued ±
1. Thus, either ω ≡
1, or ω ≡ − Example . Let us show that if F is a 1-independent NSCF over a connected space X , and moreover F is a Hilbert space, then any unitary MO on F is a scalar multipleof the identity. Let ω : X → C be such that M ω is unitary on F . Then M ∗ ω is anisometry on F ∗ , from where | ω | ≡ h x F , y F i = h M ∗ ω x F , M ∗ ω y F i = h ω ( x ) x F , ω ( y ) y F i = ω ( x ) ω ( y ) h x F , y F i . If additionally h x F , y F i 6 = 0, then ω ( x ) ω ( y ) = 1 = ω ( y ) ω ( y ), and so ω ( x ) = ω ( y ).From part (i) of Theorem 2.1 and reflexivity of F it follows that κ F is a weaklycontinuous map from X into F ∗ . Let x ∈ X . Since 0 < k x F k = h x F , x F i , thereis an open neighborhood U of x such that h x F , y F i 6 = 0 for every y ∈ U . Hence, ω ( x ) = ω ( y ), and so ω is a constant on U . Since x and U were chosen arbitrarilywe get that ω is locally a constant, and since X is connected, we conclude that ω isa constant function.The examples above suggest that the connectedness of X is a natural restrictionin the context of our investigation. Indeed, it is easy to construct counterexam-ples for disconnected spaces. Namely, let F and E be 1-independent NSCF’s overtopological spaces X and Y . Let Z be the disjoint sum of X and Y and let H = { h : Z → C , h | X ∈ F , h | Y ∈ E } endowed with a norm k h k = p k h | X k + k h | Y k .It is easy to see that H is a 1-independent NSCF over Z and a nonconstant function ω = X − Y gives rise to a unitary MO on H .On the other hand, there are naturally occurring NSCF’s on connected spaceswhich admit nontrivial unitary MO’s. Indeed, for any topological space X theoperator M ω is unitary on the NSCF C ∞ ( X ), for any ω ∈ C ( X, T ).et us analyse Example 3.3. The proof of the rigidity in that example relieson two ingredients: the different eigenspaces of an isometry are orthogonal and thepoint evaluations of two points which are “close” cannot be orthogonal. It turns outthat there is a concept of orthogonality in the general normed spaces that can beutilized to the same effect.Let E be a normed space. A vector e ∈ E is called Birkhoff (or Birkhoff-James)orthogonal to f ∈ E , if k e k ≤ k e + tf k for any t ∈ C , i.e. k e k = k P e k , where P isthe quotient map from E onto E/ span { f } . If E is a Hilbert space, then P is theorthogonal projection onto E ⊖ span f , and so the notion of Birkhoff orthogonalitycoincides with the usual one. Note however, that in general the Birkhoff orthogo-nality is NOT a symmetric relation, which is one of the crucial differences betweenthese concepts. This inspired our notation e ⊢ f for “ e is Birkhoff orthogonal to f ”. There are other generalizations of the notion of orthogonality, some of whichare symmetric, but we will only use the Birkhoff orthogonality. More details on thesubject can be found e.g. in [4] or [13, Section 1.4]. The following lemma shows thatdifferent eigenspaces of an isometry on a normed space are Birkhoff orthogonal. Lemma 3.4.
Let E be a normed space and let T : E → E be an isometry. If e, f ∈ E \ { E } are such that T e = αe and T f = βf , for some distinct α, β ∈ C ,then e ⊢ f and f ⊢ e .Proof. Let γ = βα = 1, and so T f = γαf . Since T is an isometry, it follows that α, β, γ ∈ T , and also k f + γte k = k T f + γtT e k = k γαf + γαte k = k f + te k , for any t ∈ C . Applying this equality n times we get that k f + γ n te k = k f + te k ,for any n ∈ N . Since γ = 1 the set { γ n t, n ∈ N } is either a regular polygon centeredat 0, or a dense subset of t T , and so its convex hull contains 0. Hence, from theconvexity of the function t → k f + te k , we get that k f k ≤ k f + te k , for any t ∈ C ,i.e. f ⊢ e . Due to symmetry, e ⊢ f .Let F be a 1-independent NSCF over a Hausdorff space X . Let us introducea graph structure generated by F . The Birkhoff graph of F is the graph with X serving as a set of vertices, and x, y ∈ X are joined with an edge if either x F y F ,or y F x F . The connected components of Y ⊂ X in this graph are the classes ofthe minimal equivalence relation on Y which includes all pairs ( x, y ) ∈ Y × Y suchthat x F y F . Now we can state the criterion of the rigidity in terms of the Birkhoffgraph. Proposition 3.5.
Let F be a -independent NSCF over a Hausdorff space X . Let Y ⊂ X be connected in the Birkhoff graph of F . Let T : span κ F ( Y ) → span κ F ( Y ) be a linear isometry such that y F is an eigenvector of T , for every y ∈ Y . Then T is a scalar multiple of the identity.Proof. Define ω : Y → C by T y F = ω ( y ) y F , for y ∈ Y . Let “ ∼ ” be a relation on Y defined by x ∼ y if ω ( x ) = ω ( y ). It is clear that this is an equivalence relation.It follows from Lemma 3.4 that x F y F ⇒ x ∼ y . Hence, ∼ is an equivalencerelation that contains all pairs ( x, y ) ∈ Y × Y such that x F y F , and so its classesf equivalence should contain the connected components of Y in the Birkhoff graphof F . Since Y is connected in that graph, it follows that ω ( x ) = ω ( y ), for every x, y ∈ Y . Thus, ω ≡ λ , for some λ ∈ T , and so T = λId span κ F ( Y ) . Corollary 3.6.
Let F be a -independent NSCF over a Hausdorff space X such thatthe Birkhoff graph of F is connected. If ω : Y → C is such that M ω is a unitary on F , then ω ≡ λ , where λ ∈ T . In the light of the corollary above we have to find sufficient conditions for a NSCFto have a connected Birkhoff graph. It is natural to expect that the connectednessof the phase space plays a role. In order to extend the proof from Example 3.3 tothe general case we have to find out how far can we push “nearby points cannothave orthogonal point evaluations” argument. For this we need some additionalinformation about Birkhoff orthogonality (see more in the next section).Let E be a normed space. For e ∈ E { E } let e k = (cid:8) ν ∈ B E ∗ , h e, ν i = k e k (cid:9) , i.e. e k = B E ∗ T e − ( k e k ), where e is viewed as a functional on E ∗ . This set is closedand convex, and it is easy to see that it is in fact included in ∂B E ∗ . It is well-knownthat e ⊢ f if and only if e k ∩ f ⊥ = ∅ , where f ⊥ ⊂ E ∗ . Indeed, if P : E → E/ span f is a quotient map, then P ∗ is the isometry from ( E/ span f ) ∗ into f ⊥ (see the proofof [11, Proposition 2.6]), and so k P e k = sup ν ∈ f ⊥ T B E ∗ |h e, ν i| . Hence, from the weak*compactness of the balanced set f ⊥ T B E ∗ and weak* continuity of e it follows that k P e k = k e k if and only if there exists ν ∈ e k ∩ f ⊥ . Using this information we canstate the second ingredient of our main result. Proposition 3.7.
Let F be a weakly compactly embedded -independent NSCF overa Hausdorff space X . Let x ∈ X be such that the set n f ∈ B F C ( X ) | f ( x ) = k x F k o is equicontinuous. Then the closed neighborhood of x in the Birkhoff graph of F isa neighborhood of x in X .Proof. Let E = span κ F ( X ) ⊂ F ∗ . Since from Theorem 2.3 the closed unit ballof E ∗ is B F C ( X ) , it follows that x k F = n f ∈ B F C ( X ) | f ( x ) = k x F k o . Since this set isequicontinuous, there is an open neighborhood U of x such that | f ( y ) − f ( x ) | ≤ k x F k , and so f ( y ) = 0, for every y ∈ U and f ∈ x k F . Hence, x F y F , for every y ∈ U , and so U is contained in the closed neighborhood of x in the Birkhoff graphof F . Corollary 3.8.
Let F be a weakly compactly embedded -independent NSCF over X . Then every connected Y ⊂ X is connected in the Birkhoff graph of F , if one ofthe following conditions is satisfied:(i) For any x ∈ X the set n f ∈ B F C ( X ) | f ( x ) = k x F k o is equicontinuous;(ii) For any x ∈ X the set n f ∈ B F C ( X ) | f ( x ) = k x F k o is finitely dimensional; Recall that a neighborhood of a vertex x in the graph is the set of all vertices joined with x ,while a closed neighborhood of x is the union of the neighborhood of x and { x } . iii) X is compactly generated, and for any x ∈ X the set n f ∈ B F C ( X ) | f ( x ) = k x F k o is compact.Proof. If (i) is satisfied, then the components of the Birkhoff graph of F are disjointand open, due to Proposition 3.7. Hence, every connected subset of X is completelyincluded in one of these components, and so is graph-connected.At the same time, (iii) implies (i) by virtue of Arzela-Ascoli theorem. Moreover,(ii) also implies (i) since every bounded finitely dimensional set is always equicon-tinuous. Indeed, such set is contained in a convex hull of a finite set. Since a finiteset of functions is always equicontinuous, and a convex hull of an equicontinuous setis equicontinuous, the implication follows.Thus, if the conditions of the corollary above are fulfilled and X is connected, theonly unitary MO’s on F are the scalar multiples of the identity, by virtue of Corollary3.6. However, these conditions can be difficult to check, and so it is desirable to findstronger conditions which are more readily verifiable. It turns out that one suchcondition is of geometric nature. A normed space F is called nearly strictly convex if the convex subsets of the unit sphere ∂B F are precompact (i.e. totally bounded)in F . Note that if F is a Banach space, then it is nearly strictly convex if and onlyif the closed convex subsets of ∂B F are compact. It is clear that finitely dimensionalnormed spaces are nearly strictly convex, as well as strictly- and uniformly convexnormed spaces, including Hilbert spaces and taking L p spaces, for p ∈ (1 , + ∞ ) (see[11, Definition 7.6, Definition 9.1 and Theorem 9.3]). Furthermore, a linear subspaceof a nearly strictly convex normed space is nearly strictly convex. Also, this class ofnormed spaces is closed under l p sums, for p ∈ (1 , + ∞ ) (see [22] and also Remark4.5). We can now state our main results. Theorem 3.9.
Let F be a -independent NSCF over a connected compactly gener-ated space X . If ω : Y → C is such that M ω is unitary on F then ω ≡ λ , for some λ ∈ T , provided that one of the following conditions is satisfied:(i) F is compactly embedded;(ii) F is weakly compactly embedded and nearly strictly convex, and B F is closed in C ( X ) ;(iii) F is weakly compactly embedded and F ∗∗ is nearly strictly convex;(iv) F is reflexive and nearly strictly convex.Proof. Let x ∈ X be arbitrary. In the light of Corollary 3.6 and the condition (iii)of Corollary 3.8 it is enough to show that each of the conditions (i)-(iv) imply thatthe set L x = n f ∈ B F C ( X ) | f ( x ) = k x F k o is compact in C ( X ).If (i) holds, then B F C ( X ) is compact, and so is its closed subset L x .Assume that (ii) holds. Then B F = B F C ( X ) , and so L x is a closed convex subsetof ∂B F . Since F is nearly strictly convex it follows that L x is compact in F . Sincethe topology of F is stronger than the compact-open topology, we conclude that L x is compact in C ( X ).If (iii) holds, then since B F C ( X ) = J ∗∗ F B F ∗∗ , we have that L x is the image under J ∗∗ F of the set N x = (cid:8) g ∈ B F ∗∗ |h g, x F i = k x F k (cid:9) . Clearly, N x ⊂ ∂B F ∗∗ and is aonvex set. Since F ∗∗ is nearly strictly convex, it follows that N x is compact. Hence,as J ∗∗ F is continuous from F ∗∗ into C ( X ), it follows that L x is also compact.Finally, observe that (iv) implies (iii). Indeed, every reflexive NSCF is weaklycompactly embedded, and if F is reflexive and nearly strictly convex, then F ∗∗ = F is nearly strictly convex. Remark . Note that the condition (iv) is only imposed on the Banach spaceproperties of F and has nothing to do with its embedding into C ( X ). Remark . It is desirable to relax the conditions of the theorem. In fact, at themoment we do not have an example of a non-trivial unitary MO on an either weaklycompactly embedded or nearly strictly convex NSCF over a connected compactlygenerated space.The statement can be adjusted to get rid of the 1-independence.
Proposition 3.12.
Let F be a NSCF over a Hausdorff space X such that the set { x ∈ X | x F = 0 F ∗ } is connected and one of the conditions of Theorem 3.9 are met.Then every unitary MO on F is a scalar multiple of Id F . Let us consider an example of a NSCF over a disconnected space whose Birkhoffgraph is connected nonetheless.
Example . Let (
X, d ) and z ∈ X be as in Example 2.6. Additionally assume thatthe distances between components of X is less than 1. We will show that the Birkhoffgraph of F = Lip ( X, d ) is connected. Let x ∈ X . Since the distance betweencomponents containing x and z is less than 1, there are y in the component of x and w in the component of z such that d ( y, w ) <
1. Then k w F k = max { , d ( w, z ) } >d ( w, y ) = k w F + ( − y F k , and so w F y F . Due to Corollary 3.8 there are pathsfrom x to y and from w to z in the Birkhoff graph, while y and w are joined withan edge. Hence, there is a path from x to z , and since x was chosen arbitrarily, weconclude that the Birkhoff graph of F is connected. Thus, due to Proposition 3.6,the only unitary MO’s on F are the scalar multiples of the identity.Similarly to Theorem 3.9, we can prove an analogous statement for WCO’s. Proposition 3.14.
Let F be a -independent NSCF over a Hausdorff space X thatsatisfies one of the conditions of Theorem 3.9. Let E be a NSCF over a Hausdorffspace Y . If Φ : Y → X is such that Φ ( Y ) is connected, and ω, υ : Y → C \ { } aresuch that there is a unitary S : F → F such that W Φ ,ω = W ϕ,υ S (e.g. if both W Φ ,ω and W Φ ,υ are unitaries), then υ = λω , for some λ ∈ T .Proof. First, note that S ∗ is an isometry such that S ∗ Φ ( y ) F = ω ( y ) υ ( y ) Φ ( y ) F , for every y ∈ Y . Since Φ ( Y ) is connected, the result is obtained by combining Proposition3.5 with Corollary 3.8. Remark . It is clear that Φ ( Y ) is connected in the case when Y is connectedand Φ is continuous, and also in the case when X is connected and Φ is a surjection.Moreover, continuity of Φ often holds automatically for WCO’s between NSCF’s (see[7, Corollary 3.3, Theorem 3.10 and Theorem 3.12]), while surjectivity of Φ also canbe deduced from the properties of the WCO (see in [7, Proposition 2.11]). In fact,f X is a manifold, F is 2-independent with x → k x F k continuous, lim ∞ k x F k = + ∞ and bounded functions form a dense subset of F , then F is rigid in the followingstronger sense: if Φ : X → X and ω, υ : X → C are such that W Φ ,ω and W Φ ,υ areunitaries, then Φ is a self-homeomorphism of X and ω = λυ , for some λ ∈ T , arecontinuous and non-vanishing.Up to this point in this section the word “unitary” could be replaced with theword “co-isometry”. Note however, that due to part (iii) of Corollary 2.13, any MObetween complete NSCF’s, which is a co-isometry is automatically a unitary. Let usconclude the section with a version of Theorem 3.9 for non-surjective isometries.
Proposition 3.16.
Let F be a -independent NSCF over a Hausdorff space X suchthat one of the conditions of Theorem 3.9 are met. Let ω : X → C \ { } be such that M ω is an isometry on F . Assume that Y is a dense connected subset of X such thatfor every x ∈ Y there is f ∈ F such that f ( x ) = 0 and ω − n f ∈ F , for every n ∈ N .Then ω ≡ λ , for some λ ∈ T .Proof. First, note that ω is continuous by virtue of part (i) of Proposition 2.9, andsince it does not vanish, ω is also continuous.Let E = T n ∈ N M nω F , which is a closed subspace of F . Then E is a NSCF over X that satisfies conditions of Theorem 3.9. Indeed, a closed subspace of a (weakly)compactly embedded NSCF is also a a (weakly) compactly embedded NSCF; a closedsubspace of nearly strictly convex or a reflexive normed space is nearly strictly convexor reflexive; if F ∗∗ is nearly strictly convex, then so is E ∗∗ ⊂ F ∗∗ ; finally, if B F isclosed in C ( X ), then so is B E = T n ∈ N M nω B F , as M ω is a self-homeomorphism of C ( X ).The set Z = { x ∈ X | x E = 0 E ∗ } contains Y . Indeed, for every x ∈ Y there is f ∈ F such that f ( x ) = 0 and f ∈ M nω F , for every n ∈ N . Hence, Z is connectedand dense in X . Since M ω is a unitary on E , by Proposition 3.12, it follows that ω is a constant function on Z . As Z is dense in X and ω is continuous, the resultfollows. In this section we gather some leftover results and remarks that are not directlyrelated to NSCF’s, and instead are given in the context of abstract normed spaces.Let us start by revisiting one of intuitive aspects of the orthogonality in the innerproduct spaces. Namely, one can view orthogonal vectors as “separated”. Moreprecisely, for any e = 0 E in a Hilbert space E , e ⊥ is a hyperplane, which is a closedconvex (and so weakly closed) set not containing e . It is natural to ask whether thesame phenomenon holds in general normed spaces.As was already mentioned, the relation ⊢ of Birkhoff orthogonality is not sym-metric in general normed spaces. Hence, if E is a normed space and e = 0 E ,we can consider distinct orthogonal complements e ⊢ = { f ∈ E | e ⊢ f } and ⊢ e = An operator between normed spaces is called a co-isometry if its adjoint is an isometry. f ∈ E | f ⊢ e } . From the characterization of Birkhoff orthogonality, ⊢ e is the set ofall maximal elements of functionals in e ⊥ ⊂ E ∗ , while e ⊢ = S ν ∈ e k ν ⊥ . It is easy to seethat the set { ( e, f ) ∈ E × E | e ⊢ f } is norm-closed in E × E , and so both e ⊢ and ⊢ e are closed with respect to the norm topology on E . However we cannot immediatelyconclude that these sets are weakly closed since they are usually not convex. Morespecifically, e ⊢ is a union of hyperplanes. It turns out that the key factor in thequestion of when this set is weakly convex is how “many” hyperplanes are involved. Theorem 4.1.
For a nonzero vector e ∈ E the following are equivalent:(i) e ⊢ is weakly closed;(ii) e is weakly separated from e ⊢ (i.e. e does not belongs to the weak closure of e ⊢ );(iii) e ⊢ is not weakly dense in E ;(iv) The set e k is of finite-dimension.Proof. (i) ⇒ (ii) ⇒ (iii) is trivial. Let us prove (iii) ⇒ (iv): Assume, there are g ∈ E and { ν , ..., ν n } ⊂ E ∗ , such that V = (cid:8) f ∈ E (cid:12)(cid:12) ∀ j ∈ , n |h f − g, ν j i| < (cid:9) does notintersect e ⊢ . Take a nonzero f ∈ { ν , ..., ν n } ⊥ . Then h g + tf − g, ν j i = 0, for all j ∈ , n , and so g + tf ∈ V , for any t ∈ C . For any ν ∈ e k we have that h g + tf, ν i = h g, ν i + t h f, ν i . If h f, ν i 6 = 0, for t = − h g,ν ih f,ν i we have that h g + tf, ν i = 0, whichcontradicts the assumption V T e ⊢ = ∅ . Hence ν ∈ f ⊥ , and from the arbitrarinessof f and ν , we get that e k ⊂ (cid:16) { ν , ..., ν n } ⊥ (cid:17) ⊥ = span { ν , ..., ν n } .(iv) ⇒ (i): Assume that e k is finite-dimensional. Since this set is bounded, thereis a finite collection D = { ν , ν , ..., ν n } ⊂ E ∗ , such that e k ⊂ conv D . Let g e ⊢ .Then h g, ν i 6 = 0, for any ν ∈ e k , and due to weak* compactness of D and continuityof g as a functional on D , there is δ > |h g, ν i| ≥ δ , for any ν ∈ e k .The set U = (cid:8) f ∈ E (cid:12)(cid:12) ∀ j ∈ , n |h f − g, ν j i| < δ (cid:9) is a weakly open neighborhood of g , which is disjoint from e ⊢ . Indeed, for any ν ∈ e k there are t , ..., t n , such that n P j =1 t j = 1 and ν = n P j =1 t j ν j . Then for any f ∈ U we have that |h f, ν i| = |h g, ν i + h f − g, ν i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h g, ν i + n X j =1 t j h f − g, ν j i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ |h g, ν i| − n X j =1 t j |h f − g, ν j i| > δ − n X j =1 t j δ = 0 , and so f e ⊢ . Thus, g is weakly separated from e ⊢ , and since g was chosenarbitrarily we conclude that e ⊢ is weakly closed. Remark . It would also be interesting to find necessary and sufficient conditionsfor weak closeness ⊢ e .Let us now state an interpretation of Theorem 3.9 in the context of abstractnormed spaces. heorem 4.3. Let E be a normed space and let T : E → E be an isometry. Let D ⊂ E \ { E } consist of eigenvectors of T such that span D = E . Then T = λId E ,for some λ ∈ T provided that one of the following conditions is satisfied:(i) D is connected in the norm topology;(ii) D is weakly connected and for each e ∈ D the set (cid:8) ν ∈ B E ∗ , h e, ν i = k e k (cid:9) is offinite dimension;(iii) D is bounded and weakly connected and E ∗ is separable and nearly strictly con-vex.Proof. We will only show the sufficiency of (iii). The sufficiency of (i) and (ii)is shown similarly. We can view elements of E ∗ as a NSCF over ( D, weak ). Sincespan κ E ∗ = E , it follows from Corollary 2.4 that E ∗ is a weakly compactly embedded,and also that B E ∗ is closed in C ( D ).Since E ∗ is separable it follows that a bounded set D is weakly metrizable (see[11, Proposition 3.106]). Hence, E ∗ is a nearly strictly convex NSCF over a connectedmetrizable space D . Thus, E ∗ satisfies the condition (i) of Theorem 3.9, and so itsBirkhoff graph is connected. By virtue of Proposition 3.5 we conclude that T is aconstant multiple of the identity.Let us conclude the article with discussing nearly strictly convex normed spaces.A lot of facts about strictly convex normed spaces have analogues for the nearlystrictly convex case. For example, it is easy to see that if T is a linear map from anearly strictly convex normed space E into a normed space F such that T B E = B F ,then F is also nearly strictly convex. Consequently, if H is a subspace of E which isa reflexive Banach space, then E/H is nearly strictly convex (for the proof of the factthat the quotient map maps B E onto B E/H see the proof of [17, Theorem 2.2.5]).Note that reflexivity of H is essential since any Banach space can be obtained asa quotient of a strictly convex space (see [17, Theorem 2.2.7]). Now let us discusswhen the sum of nearly strictly convex normed spaces is nearly strictly convex. Westart with a finite sum (we omit the proof in favour of the infinite case). Proposition 4.4.
Let ρ be a strictly convex norm on R n which is invariant withrespect to the the reflection over the coordinate hyperplanes. Let E , ..., E n be nearlystrictly convex normed spaces. Then E × ... × E n is nearly strictly convex withrespect to the norm k ( e , ..., e n ) k = ρ ( k e k E , ..., k e n k E n ) , ( e , ..., e n ) ∈ E × ... × E n . The analogous statement for the case of the infinite sum is more involved.
Proposition 4.5.
Let ρ : [0 , + ∞ ) N → [0 , + ∞ ] be a functional that satisfies thefollowing conditions: • ρ (0 R N ) = 0 ; ρ ( { , ..., , , , ... } ) < + ∞ ; • Positive homogeneity: ρ ( λu ) = λρ ( u ) , for any u ∈ [0 , + ∞ ) N and λ > ; • Strict subadditivity: ρ ( u + v ) ≤ ρ ( u ) + ρ ( v ) , for any u, v ∈ [0 , + ∞ ) N ; if ρ ( u + v ) = ρ ( u ) + ρ ( v ) then either v = λu , for some λ ≥ , or u = 0 R N ; • Monotonicity: ρ ( u + v ) ≥ ρ ( u ) , for any u, v ∈ [0 , + ∞ ) N ; Absolute continuity: if ρ (cid:0) { u n } n ∈ N (cid:1) < + ∞ , for some { u n } n ∈ N ∈ [0 , + ∞ ) N ,then ρ ( { , ..., , u n , u n +1 , ... } ) → , n → ∞ .Let { E n } n ∈ N be a sequence of nearly strictly convex normed spaces. Define k · k : Q n ∈ N E n → [0 , + ∞ ] by (cid:13)(cid:13) { e n } n ∈ N (cid:13)(cid:13) = ρ (cid:0) {k e n k E n } n ∈ N (cid:1) . Then E = (cid:26) e ∈ Q n ∈ N E n , k e k < + ∞ (cid:27) with the norm k · k is a nearly strictly convex normed space.Proof. For n ∈ N let E n be the completion of E n . Let (cid:16) ˜ E, k · k (cid:17) be a normed space,constructed from (cid:8) E n (cid:9) n ∈ N analogously to construction of E . We leave it to thereader to verify that E and ˜ E are linear spaces, k · k is a norm on ˜ E , and E is asubspace of ˜ E . Let us prove that E is nearly strictly convex.First, using arguments similar to the proof of [17, Theorem 2.2.1], one can showthat if ∅ = D ⊂ ∂B E is convex, and D n is the image of D under the natural pro-jection from E onto E n , then D n is a convex subset of a sphere in E n . Let r n be theradius of that sphere. For any e ∈ D we have that k e k = ρ (cid:0) { r n } n ∈ N (cid:1) = 1, and so forany f ∈ Q n ∈ N r n ∂B E n we get k f k = ρ (cid:0) { r n } n ∈ N (cid:1) = 1. Hence, B = Q n ∈ N r n ∂B E n ⊂ ∂B ˜ E .Let us show that the norm topology on B is weaker than the product topology. Let e = { e n } n ∈ N ∈ B and let ε >
0. Since ρ ( { , , ..., , r n , r n +1 , ... } ) → n → ∞ , thereis m ∈ N such that ρ ( { , , ..., , r m , r m +1 , ... } ) < ε . Let c n = ρ ( { , ..., , , , ... } ) < + ∞ , where the 1 is on the n -th position. Then, for f = { f n } n ∈ N ∈ B such that k e n − f n k E n < ε m max { ,c ,...,c m } , for every n ∈ , m , we have k e − f k ≤ m X n =1 c n k e n − f n k E n + k { , , ..., , e m , e m +1 , ... } k + k { , , ..., , f m , f m +1 , ... } k < m X n =1 c n ε m max { , c , ..., c m } + 2 ρ ( { , , ..., , r m , r m +1 , ... } ) ≤ ε ε ε. Since e and ε were chosen arbitrarily, we conclude that k · k induces a topology on B weaker than the product topology. For every n ∈ N , since E n is nearly strictlyconvex, it follows that D n is precompact in E n . Then the closure D n of D n in E n is compact. Let D ′ = Q n ∈ N D n ⊂ B , which is a compact set in the product topology,and so is compact in B . Since D ⊂ D ′ we conclude that D is relatively compact in˜ E , and so precompact in E , and so E is nearly strictly convex.Consider an example of a nearly strictly convex Banach space whose spherescontain infinite dimensional convex sets. Example . Let F = L n ∈ N l ∞ , be the l direct sum of infinite number of copiesof C with the l ∞ norm. By virtue of Proposition 4.5 this normed space is nearlystrictly convex. Let D n = (cid:8) n ⊕ t (cid:12)(cid:12) t ∈ (cid:2) − n , n (cid:3) (cid:9) be a convex subset of a spherein l ∞ of radius n . From the proof of Proposition 4.5 it follows that Q n ∈ N D n is aninfinite-dimensional convex subset of a sphere in F .Now consider an example of a non-strictly convex Banach space, such that theconvex subsets of its unit sphere are at most one-dimensional. xample . Let H be a Hilbert space, and let E = H ⊕ C . Assume that e, f ∈ H and a, b ∈ C are such that k e k + | a | = k f k + | b | = (cid:13)(cid:13) e + f (cid:13)(cid:13) + (cid:12)(cid:12) a + b (cid:12)(cid:12) = 1. Without loss ofgenerality we may assume that e = 0 H . Due to strict convexity of H there are α, β ≥ f = αe and b = βa (or else a = 0). Since we also have k f k + | b | = 1, itfollows that the convex subsets of the unit sphere that contain e ⊕ a are containedin n αe ⊕ − α k e k| a | a (cid:12)(cid:12)(cid:12) α ∈ h , k e k i o , when a = 0, and { (1 − | γ | ) e ⊕ γ, | γ | ≤ } , when a = 0.Note, that the dual E ∗ = H ⊕ ∞ C satisfies the conditions of Theorem 4.1, andso right Birkhoff orthogonal complements are weakly closed in E ∗ . Remark . It is clear that having a nearly strictly convex subset of finite co-dimension does not imply nearly strictly convexity. Indeed, even if E is a Hilbertspace, E ⊕ ∞ C is not nearly strictly convex. However, one can ask whether it is truethat if E is quasi-reflexive (i.e. such that dim E ∗∗ /E < + ∞ ) and nearly strictlyconvex, then E ∗∗ is also nearly strictly convex.Also, it is interesting whether nearly strict convexity of a normed space impliesnearly strictly convexity of its completion. Furthermore, one can study a propertystronger than nearly strictly convexity: instead of precompactness of closed convexsubsets of the unit sphere we can demand compactness. Clearly, the two conditionsare equivalent in the event when the normed space is complete.Finally, one can ask whether it is true that if E is nearly strictly convex, thenthere is a strictly convex subspace of E of finite codimension. References [1] Alexandru Aleman, Peter Duren, Mar´ıa J. Mart´ın, and Dragan Vukoti´c,
Multiplicative isome-tries and isometric zero-divisors , Canad. J. Math. (2010), no. 5, 961–974.[2] Robert F. Allen and Flavia Colonna, Multiplication operators on the Bloch space of boundedhomogeneous domains , Comput. Methods Funct. Theory (2009), no. 2, 679–693.[3] , Weighted composition operators on the Bloch space of a bounded homogeneous domain ,Topics in operator theory. Volume 1. Operators, matrices and analytic functions, Oper. TheoryAdv. Appl., vol. 202, Birkh¨auser Verlag, Basel, 2010, pp. 11–37.[4] Javier Alonso, Horst Martini, and Senlin Wu,
On Birkhoff orthogonality and isosceles orthog-onality in normed linear spaces , Aequationes Math. (2012), no. 1-2, 153–189.[5] Richard F. Arens and James Eells Jr., On embedding uniform and topological spaces , PacificJ. Math. (1956).[6] Robert G. Bartle, On compactness in functional analysis , Trans. Amer. Math. Soc. (1955),35–57.[7] Eugene Bilokopytov, Continuity and Holomorphicity of Symbols of Weighted Composition Op-erators , to appear in Complex Analysis and Operator Theory (2018).[8] Paul S. Bourdon and Sivaram K. Narayan,
Normal weighted composition operators on theHardy space H ( U ), J. Math. Anal. Appl. (2010), no. 1, 278–286.[9] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory , With the assis-tance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, IntersciencePublishers, Inc., New York; Interscience Publishers, Ltd., London, 1958.[10] Ryszard Engelking,
General topology , Second, Sigma Series in Pure Mathematics, vol. 6, Hel-dermann Verlag, Berlin, 1989. Translated from the Polish by the author.11] Mari´an Fabian, Petr Habala, Petr H´ajek, Vicente Montesinos, and V´aclav Zizler,
Banach spacetheory , CMS Books in Mathematics/Ouvrages de Math´ematiques de la SMC, Springer, NewYork, 2011. The basis for linear and nonlinear analysis.[12] J. C. Ferreira and V. A. Menegatto,
Positive definiteness, reproducing kernel Hilbert spacesand beyond , Ann. Funct. Anal. (2013), no. 1, 64–88.[13] Richard J. Fleming and James E. Jamison, Isometries on Banach spaces: function spaces ,Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 129,Chapman & Hall/CRC, Boca Raton, FL, 2003.[14] Klaus Floret,
Weakly compact sets , Lecture Notes in Mathematics, vol. 801, Springer, Berlin,1980. Lectures held at S.U.N.Y., Buffalo, in Spring 1978.[15] M. Isabel Garrido and Jes´us A. Jaramillo,
Variations on the Banach-Stone theorem , ExtractaMath. (2002), no. 3, 351–383. IV Course on Banach Spaces and Operators (Spanish)(Laredo, 2001).[16] Alexander Grothendieck, Topological vector spaces , Gordon and Breach Science Publishers,New York-London-Paris, 1973. Translated from the French by Orlando Chaljub; Notes onMathematics and its Applications.[17] Vasile I. Istr˘at¸escu,
Strict convexity and complex strict convexity , Lecture Notes in Pure andApplied Mathematics, vol. 89, Marcel Dekker, Inc., New York, 1984. Theory and applications.[18] Trieu Le,
Self-adjoint, unitary, and normal weighted composition operators in several variables ,J. Math. Anal. Appl. (2012), no. 2, 596–607.[19] ,
Normal and isometric weighted composition operators on the Fock space , Bull. Lond.Math. Soc. (2014), no. 4, 847–856.[20] Valentin Matache, Isometric weighted composition operators , New York J. Math. (2014),711–726.[21] Vladimir G. Maz’ya and Tatyana O. Shaposhnikova, Theory of Sobolev multipliers ,Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of MathematicalSciences], vol. 337, Springer-Verlag, Berlin, 2009. With applications to differential and integraloperators.[22] Chaoxun Nan and Shoubai Song,
Nearly strict convexity and best approximation , J. Math.Res. Exposition (1997), no. 4, 479–488 (English, with English and Chinese summaries).[23] Lawrence Narici and Edward Beckenstein, Topological vector spaces , Second, Pure and AppliedMathematics (Boca Raton), vol. 296, CRC Press, Boca Raton, FL, 2011.[24] Volker Scheidemann,
Introduction to complex analysis in several variables , Birkh¨auser Verlag,Basel, 2005.[25] Junzo Wada,
Weakly compact linear operators on function spaces , Osaka Math. J. (1961),169–183.[26] Dragan Vukoti´c, Pointwise multiplication operators between Bergman spaces on simply con-nected domains , Indiana Univ. Math. J. (1999), no. 3, 793–803.[27] Liankuo Zhao, Unitary weighted composition operators on the Fock space of C n , ComplexAnal. Oper. Theory (2014), no. 2, 581–590.[28] Nina Zorboska, Unitary and Normal Weighted Composition Operators on Reproducing KernelHilbert Spaces of Holomorphic Functions , preprint (2017).[29] ,
Isometric weighted composition operators on weighted Bergman spaces , J. Math. Anal.Appl.461