A Class of Operators Similar to the Shift on H 2(G)

For a compact subset K in the complex plane, let A(K) denote the algebra of all functions continuous on K and analytic on K° and let R(K) denote the uniform closure of the rational functions with poles off K. Let G is a bounded open subset whose complement in the plane has a finite number of components. Suppose that and every function in H^∞(G) is the pointwise limit of a bounded sequence of functions in . The purpose of this paper is to characterize all subnormal operators similar to M_z, the operator of multiplication by the independent variable z on the Hardy space H²(G). We also characterize all bounded linear operators that are unitarily equivalent to M_z in the case when each of the components of G is simply connected. In particular, our similarity result extends a well-known result of W. Clary on the unit disk to multiply connected domains.     

Lifting strong commutants of unbounded subnormal operators

Various theorems on lifting strong commutants of unbounded subnormal (as well as formally subnormal) operators are proved. It is shown that the strong symmetric commutant of a closed symmetric operatorS lifts to the strong commutant of some tight selfadjoint extension ofS. Strong symmetric commutants of orthogonal sums of subnormal operators are investigated. Examples of (unbounded) irreducible subnormals, pure subnormals with rich strong symmetric commutants and cyclic subnormals with highly nontrivial strong commutants are discussed.This work was supported by the KBN grant # 2P03A 041 10.     

On the Riesz representation theorem and integral operators

We present a Riesz representation theorem in the setting of extended integration theory as introduced in [6]. The result is used to obtain boundedness theorems for integral operators in the more general setting of spaces of vector valued extended integrable functions.     

Spectral measures,II: Characterization of scalar operators

We continue the development of part I. The Riesz representation theorem is proved without assuming local convexity. This theorem is applied to give sufficient conditions for an operator (continuous or otherwise) to be spectral. A uniqueness problem is pointed out and the function calculus is extended to the case of several variables. A Radon—Nikodym theorem is proved.     

The Structure of the Closure of the Rational Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>(μ)

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K°. Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let A^q(K, μ) denote the closure of A(K) in L^q(μ). The aim of this work is to study the structure of the space A^q(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A^q(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for P^q(μ), the closure of polynomials in L^q(μ), as the special case when K is a closed disk containing the support of μ.     

Factoring operators over Hilbert-Schmidt maps and vector measures

We study the structure of Banach spaces X determined by the coincidence of nuclear maps on X with certain operator ideals involving absolutely summing maps and their relatives. With the emphasis mainly on Hilbert-space valued mappings, it is shown that the class of Hilbert—Schmidt spaces arises as a ‘solution set’ of the equation involving nuclear maps and the ideal of operators factoring through Hilbert—Schmidt maps. Among other results of this type, it is also shown that Hilbert spaces can be characterised by the equality of this latter ideal with the ideal of 2-summing maps. We shall also make use of this occasion to give an alternative proof of a famous theorem of Grothendieck using some well-known results from vector measure theory.     

Fredholm weighted composition operators

We characterize the Fredholm weighted composition operators onC(X). In particular, ifX is a set with some regular property like intervals or balls inR ⁿ , our characterization implies that a weighted composition operator is Fredholm if and only if it is invertible. This equivalence is true for weighted composition operators onL ^p (), where is a nonatomic measure (1p<).     

Bounded and compact multipliers between Bergman and Hardy spaces

This paper studies the boundedness and compactness of the coefficient multiplier operators between various Bergman spacesA ^p and Hardy spacesH ^q . Some new characterizations of the multipliers between the spaces with exponents 1 or 2 are derived which, in particular, imply a Bergman space analogue of the Paley-Rudin Theorem on sparse sequences. Hardy and Bergman spaces are shown to be linked using mixed-norm spaces, and this linkage is used to improve a known result on (A ^p ,A ²), 1<p<2.Compact (H ¹,H ²) and (A ¹,A ²) multipliers are characterized. The essential norms and spectra of some multiplier operators are computed. It is shown that forp>1 there exist bounded non-compact multiplier operators fromA ^p toA ^q if and only ifpq.     

Spectral measures,III: Densely defined spectral measures

In this paper we extend the theory of spectral measures developed in Parts I and II to the case where values are assumed in the set of discontinuous (in normed spaces „unbounded”) operators. Examples of operators in nonlocally convex spaces are given, which have densely defined measures.     

Modulation spaces and pseudodifferential operators

We use methods from time-frequency analysis to study boundedness and traceclass properties of pseudodifferential operators. As natural symbol classes, we use the modulation spaces onR ^2d , which quantify the notion of the time-frequency content of a function or distribution. We show that if a symbol lies in the modulation spaceM _,1 (R ^2d ), then the corresponding pseudodifferential operator is bounded onL ²(R ^d ) and, more generally, on the modulation spacesM _p,p (R ^d ) for 1p. If lies in the modulation spaceM _2,2 ^s (R ^2d )=L _s ^/2 (R ^2d )H ^s (R ^2d ), i.e., the intersection of a weightedL ²-space and a Sobolev space, then the corresponding operator lies in a specified Schatten class. These results hold for both the Weyl and the Kohn-Nirenberg correspondences. Using recent embedding theorems of Lipschitz and Fourier spaces into modulation spaces, we show that these results improve on the classical Calderòn-Vaillancourt boundedness theorem and on Daubechies' trace-class results.     

Generalization of the Newman-Shapiro isometry theorem and Toeplitz operators

We give a generalization of the Newman-Shapiro Isometry Theorem to the case of Hilbert space-valued entire functions, which are square-summable with respect to the Gaussian measure on ⁿ , together with some applications in the theory of Toeplitz operators with operator-valued symbols. The study of various properties (such as density of domains, cores, closedness and boundedness from below) of these operators in illustrated with many relevant examples.Research supported by KBN under grant no. 2 P03A 041 10.     

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

Contractibility of the linear group in Banach spaces of measurable operators

We show contractibility to a point of the linear group for a wide class of symmetric spaces of measurable operators affiliated with several concrete non-atomic semifinite von Neumann algebras.Research supported by the Australian Research Council     

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.     

Strongly omnipresent integral operators

An operatorT on the spaceH(G) of holomorphic functions on a domainG is strongly omnipresent whenever there is a residual set of functionsfH(G) such thatT f exhibits an extremely wild behaviour near the boundary. The concept of strong omnipresence was recently introduced by the first two authors. In this paper it is proved that a large class of integral operators including Volterra operators with or without a perturbation by differential operators has this property, completing earlier work about differential and antidifferential operators.The work of the first two authors has been partially supported by DGES grant PB96-1348 and the Junta de Andalucía.     

Darboux transformations of Jacobi matrices and Padé approximation

Let J be a monic Jacobi matrix associated with the Cauchy transform F of a probability measure. We construct a pair of the lower and upper triangular block matrices L and U such that J=LU and the matrix J_C=UL is a monic generalized Jacobi matrix associated with the function F_C(λ)=λF(λ)+1. It turns out that the Christoffel transformation J_C of a bounded monic Jacobi matrix J can be unbounded. This phenomenon is shown to be related to the effect of accumulating at ∞ of the poles of the Padé approximants of the function F_C although F_C is holomorphic at ∞. The case of the UL-factorization of J is considered as well.     

Algebraic and essentially algebraic composition operators onC(X)

Summary Given a compact Hausdorff spaceX, we may associate with every continuous mapa: X X a composition operatorC _a onC(X) by the rule(C _a f)(x) = f(a(x)). We describe all self-mapsa for whichC _a is an algebraic operator or an essentially algebraic operator (i.e. an operator algebraic modulo compact operators), determine the characteristic polynomialp _a (z) and the essentially characteristic polynomialq _a (z) in these cases and show how the connectivity ofX may be characterized in terms of the quotientsp _a (z)/q _a (z). Research supported by the Alfried Krupp Förderpreis für junge Hochschullehrer.     

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {r_n(z)}_n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {r_n(z)}_n∈N in the complement CP¹?D_f, where D_f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {r_n(z)}_n∈N. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.