Showing papers in "Integral Equations and Operator Theory in 1993"

For which reproducing kernel Hilbert spaces is Pick's theorem true?

Abstract: Pick's theorem tells us that there exists a function inH∞, which is bounded by 1 and takes given values at given points, if and only if a certain matrix is positive.H∞ is the space of multipliers ofH2, and this theorem has a natural generalisation whenH∞ is replaced by the space of multipliers of a general reproducing kernel Hilbert spaceH(K) (whereK is the reproducing kernel). J. Agler has shown that this generalised theorem is true whenH(K) is a certain Sobolev space or the Dirichlet space, so it is natural to ask for which reproducing kernel Hilbert spaces this generalised theorem is true. This paper widens Agler's approach to cover reproducing kernel Hilbert spaces in general, replacing Agler's use of the deep theory of co-analytic models by a relatively elementary, and more general, matrix argument. The resulting theorem gives sufficient (and usable) conditions on the kernelK, for the generalised Pick's theorem to be true forH(K), and these are then used to prove Pick's theorem for certain weighted Hardy and Sobolev spaces and for a functional Hilbert space introduced by Saitoh.

Analytic functions of bounded mean oscillation and the Bloch space

Abstract: In this paper we show that the closure of the space BMOA of analytic functions of bounded mean oscillation in the Bloch spaceB is the image P(U) of space of all continuous functions on the maximal ideal space ofH ∞ under the Bergman projection P. It is proved that the radial growth of functions in P(U) is slower than the iterated logarithm studied by Makarov. So some geometric conditions are given for functions in P(U), which we can easily use to construct many Bloch functions not in P(U).

Strongly definitizable linear pencils in Hilbert space

Abstract: Selfadjoint linear pencils ΛF−G are considered which have discrete spectrum and neither F nor G is definite. Several characterizations are given of a “strongly definitizable” property when F and G are bounded, and also when both operators are unbounded. The theory is applied to analysis of the stability of a linear second order initial-boundary value problem with boundary conditions dependent on the eigenvalue parameter.

Fredholm weighted composition operators

Abstract: 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 (μ), where μ is a nonatomic measure (1≤p<∞).

On the Hankel-norm approximation of upper-triangular operators and matrices

Abstract: A matrix T = Tij ∞=-∞ , which consists of a doubly indexed collection Tij of operators, is said to be upper when Tij = 0 for i > j. We consider the case where the Tij are finite matrices and the operator T is bounded, and such that the Tij are generated by a strictly stable, non-stationary but linear dynamical state space model or colligation. For such a model, we consider model reduction, which is a procedure to obtain optimal approximating models of lower system order. Our approximation theory uses a norm which generalizes the Hankel norm of classical stationary linear dynamical systems. We obtain a parametrization of all solutions of the model order reduction problem in terms of a fractional representation based on a non-stationary J-unitary operator constructed from the data. In addition, we derive a state space model for the so-called maximum entropy approximant. In the stationary case, the problem was solved by Adamyan, Arov and Krein in their paper on Schur-Takagi interpolation. Our approach extends that theory to cover general, non-Toeplitz upper operators.

Extension theorems for Fredholm and invertibility symbols

Abstract: This paper is a continuation of [GK3] where the theory of Invertibility Symbol in Banach algebras was developed. In the present paper we generalize these results for the case when the Invertibility Symbol is defined on a subalgebra of the Banach algebras. The difficulty which arises here in this more general case is connected with the fact that some elements of the subalgebra may have the inverses which do not belong to the subalgebra. This generalization of the theory allows us to study the Fredholm Symbols of linear operators. Applications to subalgebras generated by two idempotents and to algebras generated by singular integral operators are presented.

Banach algebras of singular integral operators with piecewise continuous coefficients. General contour and weight

Abstract: The Banach algebra generated by one-dimensional linear singular integral operators with matrix valued piecewise continuous coefficients in the spaceLp(Γ,ρ) with an arbitrary weight ρ is studied. The contour Γ consists of a finite number of closed curves and open arcs with satisfy the Carleson condition. The contour may have a finite number of points of selfintersection. The symbol calculus in this algebra is the main result of the paper.

An a-posteriori parameter choice for simplified regularization of ill-posed problems

Abstract: An a posteriori parameter choice strategy is proposed for the simplified regularization of ill-posed problems where no information about the smoothness of the unknown solution is required. If the smoothness of the solution is known then, as a particular case, the optimal rate is achieved. Our result also includes a recent result of Guacanme (1990).

One-dimensional perturbations of the shift

Abstract: The structure of isometries on a Hilbert space are well studied. In this paper we study contractions which are one-dimensional perturbations of isometries, in particular, perturbations of the shift operator onH2.

On some algebras diagonalized by M-bases of ℓ2

Abstract: We study reflexive algebrasA whose invariant lattices LatA are generated by M-bases of l2. Examples are given whereA differs from ℱ (ℱ being the rank one subalgebra ofA), and where ℱ together with the identity I is not strongly dense inA. For M-bases in a special class, we characterize the cases when they are strong, and also when the identity I is the ultraweak limit of a sequence of contractions in ℱ. We show that this holds provided that I is approximable by compact operators inA at any two points of l2. We show that the spaceA+ℒ* (where ℒ is the annihilator of ℱ) is ultraweakly dense in ℬ(l2), and characterize the M-bases in this class for which the sum is direct. We give a class of automorphisms ofA which are strongly continuous but not spatial.

Regularized determinants for pseudodifferential operators in vector bundles overS 1

Abstract: We express the ζ-regularized determinant of an elliptic pseudodifferential operatorA overS1 with strongly invertible principal symbol in terms of the Fredholm determinant of an operator of determinant class, canonically associated toA, and local invariants. These invariants are given by explicit formulae involving the principal and subprincipal symbol of the operator. We remark that,generically, elliptic pseudodifferential operators have a strongly invertible principal symbol.

Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators

Abstract: LetA andB be anticommuting self-adjoint operators in a Hilbert space ℋ It is proven thatiAB is essentially self-adjoint on a suitable domain and its closureC(A, B) anticommutes withA andB LetU s be the partial isometry associated with the self-adjoint operatorsS, ie, the partial isometry defined by the polar decompositionS=U S |S| LetP S be the orthogonal projection onto (KerS)⊥ Then the following are proven: (i) The operatorsU A ,U B ,U C(A,B) ,P A ,P B , andP A P B multiplied by some constants satisfy a set of commutation relations, which may be regarded as an extension of that satisfied by the standard basis of the Lie algebra $$\mathfrak{s}\mathfrak{u}(2,\mathbb{C})$$ of the special unitary groupSU(2); (ii) There exists a Lie algebra $$\mathfrak{M}$$ associated with those operators; (iii) If ℋ is separable andA andB are injective, then $$\mathfrak{M}$$ gives a completely reducible representation of $$\mathfrak{s}\mathfrak{u}(2,\mathbb{C})$$ with each irreducible component being the spin representation of the Clifford algebra associated with ℝ3; This result can be extended to the case whereA andB are not necessarily injective Moreover, some properties ofA+B are discussed The abstract results are applied to Dirac operators

Semigroups of isometries, Toeplitz algebras and twisted crossed products

Abstract: We study theC*-algebras generated by projective isometric representations of semigroups, using a dilation theorem and the stucture theory of twisted crossed products. These algebras include the Toeplitz algebras of noncommutative tori recently studied by Ji, and similar algebras associated to the twisted group algebras of other groups such as the integer Heisenberg group.

Factorization of a selfadjoint nonanalytic operator function II. Single spectral zone

Abstract: We consider a selfadjoint and smooth enough operator-valued functionL(λ) on the segment [a, b]. LetL(a)≪0,L(b)≫0, and there exist two positive numbers e and δ such that the inequality |(L(λ)f, f)| δ. Then the functionL(λ) admits a factorizationL(λ)=M(λ)(λI-Z) whereM(λ) is a continuous and invertible on [a, b] operator-valued function, and operatorZ is similar to a selfadjoint one. This result was obtained in the first part of the present paper [10] under a stronge conditionL′(λ)≫0 (λ ∈ [a,b]). For analytic functionL(λ) the result of this paper was obtained in [13].

Completion problems for j pq -inner functions. I.

Abstract: This paper studies various completion problems for a subclass ofj pq-inner functions. Special attention is drawn to so-calledA-normalizedj pq-elementary factors of full-rank, which are closely related to the matricial Schur problem. Finally, as an application an inverse problem for Caratheodory sequences is answered.

The space of spectral measures is a homogeneous reductive space

Abstract: We prove that the space of spectral measures on a W*-algebra is a smooth Banach manifold in a natural way and that the action of the group of invertible elements of the algebra by inner automorphisms makes it into a reductive homogeneous space. This gives a geometric structure for the set of normal operators with the same spectrum.

Convolution operators on a finite interval with periodic kernel-Fredholm property and invertibility

Abstract: Finite interval convolution operators with periodic kernel-functions are studied from the point of view of Fredholm properties and invertibility. These operators are associated with Wiener-Hopf operators with matrix-valued symbols defined on a space of functions whose domain is a contour consisting of two parallel straight-lines. For the Fredholm study a Wiener-Hopf operator is considered on a space of functions defined on a contour composed of two closed curves having a common multiple point. Invertibility of the finite interval operator is studied for a subclass of symbols related to the problem of wave diffraction by a strip grating.

Group inverses and Drazin inverses over Banach algebras

Abstract: For a matrix over a complex commutative unital Banach algebra, necessary and sufficient conditions are given for the existence of its group inverse, and more generally, its Drazin inverses. The conditions are easy to check and explicit formulas for the inverses are provided. Some properties of the inverses and an application to operator theory are discussed. This note is a continuation of an earlier work of the author.

On spectrally associated Schur functions, Arov-inner functions and a Nehari-type completion problem for Schur functions

Abstract: This paper studies a special Nehari-type completion problem for matrix-valued Schur functions. Several necessary and sufficient conditions for solvability of the problem are given. If a solution exists, then it is unique and it can be expressed explicitly by the original data.

Leading coefficients of the eigenvalues of perturbed analytic matrix functions

Abstract: This note contains some supplements to our earlier notes [LN II], [LN III], where the Newton diagram was used in order to obtain in a straightforward way information about the perturbed eigenvalues of an analytic and analytically perturbed matrix function.

Characterization of anticommutativity of self-adjoint operators in connection with Clifford algebra and applications

Abstract: A new characterization of anticommutativity of (unbounded) self-adjoint operators is presented in connection with Clifford algebra. Some consequences of the characterization and applications are discussed.

Abstract, weighted, and multidimensional Adamjan-Arov-Krein theorems, and the singular numbers of Sarason commutants

Abstract: The classical Adamjan-Arov-Krein (A-A-K) theorem relating the singular numbers of Hankel operators to best approximations of their symbols by rational functions is given an abstract version. This provides results for Hankel operators acting in weightedH2(T; μ), as well as inH2(Td), and an A-A-K type extension of Sarason's interpolation theorem. In particular, it is shown that all compact Hankel operators inH2(Td) are zero.

Completion problems forjpq-inner functions. II

Abstract: This paper is a continuation of [AFK]. The notations used there will be preserved. We will consider completion problems forjpq-inner polynomials which have a prescribed structure, namely for such matrix polynomials which have a prescribed structure, namely for such matrix polynomials which can be considered as suitably normalized resolvent matrix of an appropriate non-degenerate matricial Schur problem.