Isometric Equivalence of Certain Operators on Banach Spaces

A pair of operators on a Banach space X are isometrically equivalent if they are intertwined by a surjective isometry of X. We investigate the isometric equivalence problem for pairs of operators on specific types of Banach spaces. We study weighted shifts on symmetric sequence spaces, elementary operators acting on an ideal I of Hilbert space operators, and composition operators on the Bloch space. This last case requires an extension of known results about surjective isometries of the Bloch space.     

Operator ergodic theory for one-parameter decomposable groups

After initial treatment of the Fourier analysis and operator ergodic theory of strongly continuous decomposable one-parameter groups of operators in the Banach space setting, we show that in the setting of a super-reflexive Banach space X these groups automatically transfer from the setting of R to X the behavior of the Hilbert kernel, as well as the Fourier multiplier actions of functions of higher variation on R. These considerations furnish one-parameter groups with counterparts for the single operator theory in Berkson (2010) [4]. Since no uniform boundedness of one-parameter groups of operators is generally assumed in the present article, its results for the super-reflexive space setting go well beyond the theory of uniformly bounded one-parameter groups on UMD spaces (which was developed in Berkson et al., 1986 [13]), and in the process they expand the scope of vector-valued transference to encompass a genre of representations of R that are not uniformly bounded.     

Properties of representations of operators acting between spaces of vector-valued functions

A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued L ¹-spaces into L ^∞-spaces are kernel operators and that in fact this relation induces an isometric isomorphism between the space of such operators and the space of all bounded kernels. We extend this result to the case of spaces of vector-valued functions. A recent result due to Arendt and Thomaschewski states that the local operators acting on L ^p -spaces of functions with values in separable Banach spaces are precisely the multiplication operators. We extend this result to non-separable dual spaces. Moreover, we relate positivity and other order properties of the operators to corresponding properties of the representations.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

Little Hankel operators on the half plane

In this paper we consider a class of weighted integral operators onL ² (0, ) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.     

Continuity properties for modulation spaces, with applications to pseudo-differential calculus—I

Let M^p,q denote the modulation space with parameters p,q∈[1,∞]. If 1/p₁+1/p₂=1+1/p₀ and 1/q₁+1/q₂=1/q₀, then it is proved that . The result is used to get inclusions between modulation spaces, Besov spaces and Schatten classes in calculus of Ψdo (pseudo-differential operators), and to extend the definition of Toeplitz operators. We also discuss continuity of ambiguity functions and Ψdo in the framework of modulation spaces.     

Strongly damped wave problems: Bootstrapping and regularity of solutions

The aim of the article is to present a unified approach to the existence, uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in Banach spaces. Our results are applied next to a number of examples appearing in literature, which fall into the class of strongly damped semilinear wave equations. The present work essentially extends the results on the existence and regularity of solutions to such problems. Previously, these problems have been considered mostly within the Hilbert space setting and with the main part operators being selfadjoint. In this article we present a more general approach, involving sectorial operators in reflexive Banach spaces.     

Differences of Composition Operators between Weighted Banach Spaces of Holomorphic Functions on the Unit Polydisk

We consider differences of composition operators between given weighted Banach spaces H ^∞ _v or H ⁰ _v of analytic functions defined on the unit polydisk D ^N with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also study boundedness and compactness of the operators. This paper is an extension of [6] where the one-dimensional case is treated. Received: May 15, 2007. Revised: October 8, 2007.     

THE UNIQUENESS OF OPERATIONAL QUANTITIES IN VON NEUMANN ALGEBRAS

Abstract

Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].     

Vector-valued multiparameter singular integrals and pseudodifferential operators

We consider multiparameter singular integrals and pseudodifferential operators acting on mixed-norm Bochner spaces Lp₁,…,pN(Rn₁×?×RnN;X) where X is a UMD Banach space satisfying Pisier's property (α). These geometric conditions are shown to be necessary. We obtain a vector-valued version of a result by R. Fefferman and Stein, also providing a new, inductive proof of the original scalar-valued theorem. Then we extend a result of Bourgain on singular integrals in UMD spaces with an unconditional basis to a multiparameter situation. Finally we carry over a result of Yamazaki on pseudodifferential operators to the Bochner space setting, improving the known vector-valued results even in the one-parameter case.     

Essential spectra of spectral operators

The various essential spectra of a linear operator have been surveyed byB. Gramsch andD. Lay [4]. In this paper we characterize the essential spectra and the related quantities nullity, defect, ascent and descent of bounded spectral operators. It is shown that a number of these spectra coincide in the case of a spectral or a scalar type operator. Some results known for normal operators in Hilbert space are extended to spectral operators in Banach space.     

Generalized Anti-Wick Operators with Symbols in Distributional Sobolev spaces

Generalized Anti-Wick operators are introduced as a class of pseudodifferential operators which depend on a symbol and two different window functions. Using symbols in Sobolev spaces with negative smoothness and windows in so-called modulation spaces, we derive new conditions for the boundedness on L² of such operators and for their membership in the Schatten classes. These results extend and refine results contained in [20], [10], [5], [4], and [14].     

PROJECTIONS ON OPERATOR DUALS

Abstract

We discuss the existence of a projection with kernel K_b(E,F) ¹ (the annihilator of the quasi-compact operators) on the dual space of the space L _b,(E, F) of continous linear operators. Our results are proved in the context of Hausdorff locally convex spaces, but also provide extensions of recent results in the context of Banach spaces.     

Quasi-multipliers and algebrizations of an operator space

Let X be an operator space, let φ be a product on X, and let (X,φ) denote the algebra that one obtains. We give necessary and sufficient conditions on the bilinear mapping φ for the algebra (X,φ) to have a completely isometric representation as an algebra of operators on some Hilbert space. In particular, we give an elegant geometrical characterization of such products by using the Haagerup tensor product. Our result makes no assumptions about identities or approximate identities. Our proof is independent of the earlier result of Blecher, Ruan and Sinclair [D.P. Blecher, Z.-J. Ruan, A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1) (1990) 188-201] which solved the case when the bilinear mapping has an identity of norm one, and our result is used to give a simple direct proof of this earlier result. We also develop further the connections between quasi-multipliers of operator spaces and their representations on a Hilbert space or their embeddings in the second dual, and show that the quasi-multipliers of operator spaces defined in [M. Kaneda, V.I. Paulsen, Quasi-multipliers of operator spaces, J. Funct. Anal. 217 (2) (2004) 347-365] coincide with their C^∗-algebraic counterparts.     

Time-Frequency analysis of localization operators

We study a class of pseudodifferential operators known as time-frequency localization operators, Anti-Wick operators, Gabor-Toeplitz operators or wave packets. Given a symbol a and two windows ?₁,?₂, we investigate the multilinear mapping from to the localization operator A_a^?₁,?₂ and we give sufficient and necessary conditions for A_a^?₁,?₂ to be bounded or to belong to a Schatten class. Our results are formulated in terms of time-frequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.     

Norm-attaining compact operators

We show examples of compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. This is the negative answer to an open question posed in the 1970s. Actually, any strictly convex Banach space failing the approximation property serves as the range space. On the other hand, there are examples in which the domain space has a Schauder basis.     

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c₀ and ?_p for 1?p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E?(⊕?₂ⁿ)_c0, that is, E is the c₀-direct sum of the finite-dimensional Hilbert spaces ?₂¹,?₂²,…,?₂ⁿ,… .     

The Weiss conjecture on admissibility of observation operators for contraction semigroups

We prove the conjecture of George Weiss for contraction semigroups on Hilbert spaces, giving a characterization of infinite-time admissible observation functionals for a contraction semigroup, namely that such a functionalC is infinite-time admissible if and only if there is anM>0 such that for alls in the open right half-plane. HereA denotes the infinitesimal generator of the semigroup. The result provides a simultaneous generalization of several celebrated results from the theory of Hardy spaces involving Carleson measures and Hankel operators.     

Hilbert-Carleman and regularized determinants for linear operators

A general theory of regularized and Hilbert-Carleman determinants in normed algebras of operators acting in Banach spaces is proposed. In this approach regularized determinants are defined as continuous extensions of the corresponding determinants of finite dimensional operators. We characterize the algebras for which such extensions exist, describe the main properties of the extended determinants, obtain Cramer's rule and the formulas for the resolvent which are expressed via the extended tracestr(A ^k ) of iterations and regularized determinants.This paper is a continuation of the paper [GGKr].     

Essential norm of the difference of weighted composition operators

We study differences of weighted composition operators between weighted Banach spaces H _ν ^∞ of analytic functions with weighted sup-norms and give an expression for the essential norm of these differences. We apply our result to estimate the essential norm of differences of composition operators acting on Bloch-type spaces. Authors’ addresses: Mikael Lindstr?m, Department of Mathematics, Abo Akademi University, FIN 20500 Abo, Finland; Elke Wolf, Mathematical Institute, University of Paderborn, D-33095 Paderborn, Germany