On the Decomposition of the Riesz Operator and the Expansion of the Riesz Semigroup

For a Riesz operator T on a reflexive Banach space X with nonzero eigenvalues denote by E(λ_i; T) the eigen-projection corresponding to an eigenvalue λ_i. In this paper we will show that if the operator sequence is uniformly bounded, then the Riesz operator T can be decomposed into the sum of two operators T_p and T_r: T = T_p + T_r, where T_p is the weak limit of T_n and T_r is quasi-nilpotent. The result is used to obtain an expansion of a Riesz semigroup T(t) for t ≥ τ. As an application, we consider the solution of transport equation on a bounded convex body.     

Finite Laurent Developments and the Logarithmic Residue Theorem in the Real Non-analytic Case

This paper develops a general abstract non-holomorphic operator calculus under minimal regularity requirements on the family of operators through the concept of algebraic eigenvalue and the use of a, very recent, transversalization theory. Further, it analyzes under what conditions the inverse of a non-analytic family admits a finite Laurent development, and employs the new findings to calculate the multiplicity of a real non-analytic family through a logarithmic residue, so extending the applicability of the classical theory of I. C. Gohberg and coworkers. Applications to matrix families and Nonlinear Analysis are also explained.     

On Fredholm index in Banach spaces

We prove the stability of the index of a Fredholm complex of Banach spaces under those compact perturbations which are uniform limits of finite-rank operators (Theorem 3.3). This result is a consequence of some similar statements (Theorems 3.1 and 3.2) concerning more general objects, namely the Fredholm pairs (Definition 1.1).     

The Virozub–Matsaev Condition and Spectrum of Definite Type for Self-adjoint Operator Functions

We establish sufficient conditions for the so-called Virozub–Matsaev condition for twice continuously differentiable self-adjoint operator functions. This condition is closely related to the existence of a local spectral function and to the notion of positive type spectrum. Applications to self-adjoint operators in Krein spaces and to quadratic operator polynomials are given. Received: September 22, 2007. Accepted: September 29, 2007.     

On the Spectrum of Invertible Semi-hyponormal Operators

It is known that for a semi-hyponormal operator, the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. The original proof of this theorem involves the so-called singular integral model. The purpose of this paper is to give a different proof of the same theorem for the case of invertible semi-hyponormal operators without using the singular integral model.      

On the Isolated Points of the Spectrum of Paranormal Operators

For paranormal operator T on a separable complex Hilbert space we show that (1) Weyl’s theorem holds for T, i.e., σ(T) \ w(T) = π₀₀(T) and (2) every Riesz idempotent E with respect to a non-zero isolated point λ of σ(T) is self-adjoint (i.e., it is an orthogonal projection) and satisfies that ranE = ker(T − λ) = ker(T − λ)^*.     

Q-functions of Hermitian Contractions of Krein-Ovcharenko Type

In this paper operator-valued Q-functions of Krein-Ovcharenko type are introduced. Such functions arise from the extension theory of Hermitian contractive operators A in a Hilbert space ℌ. The definition is related to the investigations of M.G. Krein and I.E. Ovcharenko of the so-called Q_μ- and Q_M-functions. It turns out that their characterizations of such functions hold true only in the matrix valued case. The present paper extends the corresponding properties for wider classes of selfadjoint contractive extensions of A. For this purpose some peculiar but fundamental properties on the behaviour of operator ranges of positive operators will be used. Also proper characterizations for Q_μ- and Q_M-functions in the general operator-valued case are given. Shorted operators and parallel sums of positive operators will be needed to give a geometric understanding of the function-theoretic properties of the corresponding Q-functions.     

On a nonlinear eigenvalue problem

In the present paper we consider a selfadjoint and nonsmooth operator-valued function on (c, d)R ¹. We suppose that the equation (L()x, x)=0,x0, has exactly one rootp(x) (c, d) and the functionf()=(L()x, x) is increasing at the pointp(x). We discuss questions of the variational theory of the spectrum. Some theorems on the variational properties of the spectrum are proved.     

Norms of Moore-Penrose Inverses of Fredholm Toeplitz Operators

In this paper we estimate the norm of the Moore-Penrose inverse T(a)⁺ of a Fredholm Toeplitz operator T(a) with a matrix-valued symbol a∈L_{N × N}^∞ defined on the complex unit circle. In particular, we show that in the ”generic case” the strict inequality ||T(a)⁺|| > ||a⁻¹||_∞ holds. Moreover, we discuss the asymptotic behavior of ||T(t^ra)⁺|| for . The results are illustrated by numerical experiments.     

On the Range of Elementary Operators

Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Let A = (A₁,A₂,.., A_n) and B = (B₁, B₂,.., B_n) be n-tuples in B(H), we define the elementary operator by In this paper we initiate the study of some properties of the range of such operators.     

The Mosaic and Principal Function of a Subnormal Operator

In this paper, we use the mosaic of a subnormal operator given by Daoxing Xia to give an alternate definition of the Pincus principal function for pure subnormal operators. This allows us to provide much simplified proofs of some of the basic properties of the principal function and of the Carey-Helton-Howe-Pincus Theorem in the subnormal case.     

Solvability and Fredholm Properties of Integral Equations on the Half-Line in Weighted Spaces

The solvability of integral equations of the form and the behaviour of the solution x at infinity are investigated. Conditions on k and on a weight function w are obtained which ensure that the integral operator K with kernel k is bounded as an operator on X_w, where X_w denotes the weighted space of those continuous functions defined on the half-line which are O(w(s)) as We also derive conditions on w and k which imply that the spectrum and essential spectrum of K on X_w are the same as on BC[0,). In particular, the results apply when when the integral equation is of Wiener-Hopf type. In this case we show that our results are particularly sharp.     

Linearization,trace, and determinant of certain meromorphic operator functions

In this paper operator functions of type

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<).     

A structure theorem for right invariant right holoids

Communicated by Boris M. Schein     

The Fredholm Index for Elements of Toeplitz-Composition C<Superscript>*</Superscript>-Algebras

We analyze the essential sectrum and index theory of elements of Toeplitz-composition C^*-algebras (algebras generated by the Toeplitz algebra and a single linear-fractional composition operator, acting on the Hardy space of the unit disk). For automorphic composition operators we show that the quotient of the Toeplitz-composition algebra by the compacts is isomorphic to the crossed product C^*-algebra for the action of the symbol on the boundary circle. Using this result we obtain sufficient conditions for polynomial elements of the algebra to be Fredholm, by analyzing the spectrum of elements of the crossed product. We also obtain an integral formula for the Fredholm index in terms of a generalized Chern character. Finally we prove an index formula for the case of the non-parabolic, non-automorphic linear fractional maps studied by Kriete, MacCluer and Moorhouse.     

(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Greens functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants.We illustrate our approach by identifying classical objects such as the Jost function for half-line Schrödinger operators and the inverse transmission coe.cient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known expressions for them in due course. We also apply our formalism to Floquet theory of Schrödinger operators, and upon identifying the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant.Finally, we rederive the explicit formula for the 2-modified Fredholm determinant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener-Hopf analog of Days formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.     

On Smooth Local Resolvents

We exhibit an example of a bounded linear operator on a Banach space which admits an everywhere defined local resolvent with continuous derivatives of all orders.