On Selfadjoint Homogeneous Operators

We study unbounded selfadjoint operators that are unitarily equivalent to their affine transformation. We investigate transformation properties of an operator-valued M-function associated with given affine-invariant operator as well as spectral properties of operators that occur in that investigation.     

关于自伴算子的奇异连续谱

本文建立了自伴算子奇异连续谱的三个定理,它们推广了Barry Simon近期所获得的一些重要结果.     

Variation of Discrete Spectra for Non-Selfadjoint Perturbations of Selfadjoint Operators

Let B = A + K where A is a bounded selfadjoint operator and K is an element of the von Neumann–Schatten ideal ${\mathcal{S}_{p}}$ with p > 1. Let {λ _n } denote an enumeration of the discrete spectrum of B. We show that ${\sum_n {\rm dist}(\lambda_n, \sigma(A))^p}$ is bounded from above by a constant multiple of ${\|K\|_{p}^p}$ . We also derive a unitary analog of this estimate and apply it to obtain new estimates on zero-sets of Cauchy transforms.     

About the Limit Spectra of Some Unbounded Selfadjoint Integral Operators

Siberian Mathematical Journal - We establish sufficient conditions for the zero to belong to the limit spectra of some unbounded selfadjoint integral operator.     

Positive Bernstein-Sheffer Operators

Let h(t) = Σ_{n ≥ 1}h_nt_n, h₁ > 0, and exp(xh(t)) = Σ_{n ≥ 0}P_n(x) tⁿ/n!. For f C[0,1], the associated Bernstein-Sheffer operator of degree n is defined by B^h_nf(x) = P_n^{− 1} Σⁿ_{k = 0}f(k/n)(ⁿ_k) P_k(x) P_{n − k}(1 − x) where p_n = p_n(1). We characterize functions h for which B^h_n is a positive operator for all n ≥ 0. Then we give a necessary and sufficient condition insuring the uniform convergence of B^h_nf to f. When h is a polynomial, we give an upper bound for the error f − B^h_nf _∞. We also discuss the behavior of B^h_nf when h is a series with a finite or infinite radius of convergence.     

Positive Refinable Operators

Recently, linear positive operators of Bernstein–Schoenberg type, relative to B-splines bases, have been considered. The properties of these operators are derived mainly from the total positivity of normalized B-spline bases. In this paper we shall construct a generalization of the operator considered in [15] by means of normalized totally positive bases generated by a particular class of totally positive scaling functions. Next, we shall study its approximation properties. Our results can be established also for more general sequences of normalized totally positive bases.     

Invariant Maximal Positive Subspaces and Polar Decompositions

It is proved that invertible operators on a Krein space which have an invariant maximal uniformly positive subspace and map its orthogonal complement into a nonnegative subspace allow polar decompositions with additional spectral properties. As a corollary, several classes of Krein space operators are shown to allow polar decompositions. An example in a finite dimensional Krein space shows that there exist dissipative operators that do not allow polar decompositions.     

Some Selfadjoint 2$\times$2 Operator Matrices Associated with Closed Operators

We study a 22 operator matrix associated with a closed densely defined operator. Among others, the selfadjointness of a closed symmetric operator and the strong commutativity of two (unbounded) self-adjoint operators are characterized in terms of the related operator matrices. We propose a definition of strong commutativity for closed symmetric operators. Submitted: November 8, 2001     

Positive Semigroups of Kernel Operators

Extending results of Davies and of Keicher on ℓ^p we show that the peripheral point spectrum of the generator of a positive bounded C₀-semigroup of kernel operators on L^p is reduced to 0. It is shown that this implies convergence to an equilibrium if the semigroup is also irreducible and the fixed space non-trivial. The results are applied to elliptic operators. Dedicated to the memory of H.H. Schaefer     

Traceability of Positive Integral Operators

A necessary and sufficient condition is given for a positive bounded linear operator with an integral kernel to be trace class on L²(μ) for a σ-finite measure μ. The condition refines earlier criteria for positive Hilbert–Schmidt operators and positive integral operators with continuous kernels on a locally compact space.     

Positive Invertibility of Nonselfadjoint Operators

The paper deals with a class of nonselfadjoint operators in a separable Hilbert lattice. Conditions for the positive invertibility are derived. Moreover, upper and lower estimates for the inverse operator are established. In addition, bounds for the positive spectrum are suggested. Applications to integral operators, integro-differential operators and infinite matrices are discussed.     

Commuting Linear Operators and Decompositions; Applications to Einstein Manifolds

For linear operators which factor P=P ₀ P ₁ ??? P _? , with suitable assumptions concerning commutativity of the factors, we introduce several notions of a decomposition. When any of these hold then questions of null space and range are subordinated to the same questions for the factors, or certain compositions thereof. When the operators P _i are polynomial in other commuting operators \(\mathcal{D}_{1},\ldots,\mathcal{D}_{k}\) then we show that, in a suitable sense, generically factorisations algebraically yield decompositions. In the case of operators on a vector space over an algebraically closed field this boils down to elementary algebraic geometry arising from the polynomial formula for P. The results and formulae are independent of the \(\mathcal{D}_{j}\) and so the theory provides a route to studying the solution space and the inhomogenous problem Pu=f without any attempt to “diagonalise” the \(\mathcal{D}_{j}\). Applications include the construction of fundamental solutions (or “Greens functions”) for PDE; analysis of the symmetry algebra for PDE; direct decompositions of Lie group representations into Casimir generalised eigenspaces and related decompositions of vector bundle section spaces on suitable geometries. Operators P polynomial in a single other operator \(\mathcal{D}\) form the simplest case of the general development and here we give universal formulae for the projectors administering the decomposition. As a concrete geometric application, on Einstein manifolds we describe the direct decomposition of the solution space and the general inhomogeneous problem for the conformal Laplacian operators of Graham-Jenne-Mason-Sparling.     

On Finite Rank Perturbations of Selfadjoint Operators in Krein Spaces and Eigenvalues in Spectral Gaps

It is shown that the finiteness of eigenvalues in a spectral gap of a definitizable or locally definitizable selfadjoint operator in a Krein space is preserved under finite rank perturbations. This results is applied to a class of singular Sturm–Liouville operators with an indefinite weight function.