Let S be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non-zero deficiency indices. Let J be an open subset of R. If there exists a self-adjoint extension S0 of S such that J is contained in the resolvent set of S0 and the associated Weyl function of the pair {S, S0} is monotone with respect to J , then for any self-adjoint operator R there exists ...

We study weighted composition operators on Hilbert spaces of analytic functions on the unit ball with kernels of the form (1− 〈z, w〉)−γ for γ > 0. We find necessary and sufficient conditions for the adjoint of a weighted composition operator to be a weighted composition operator or the inverse of a weighted composition operator. We then obtain characterizations of self-adjoint and unitary weigh...

For a bounded non-negative self-adjoint operator acting in a complex, infinitedimensional, separable Hilbert space H and possessing a dense range R we propose a new approach to characterisation of phenomenon concerning the existence of subspaces M ⊂ H such that M ∩R = M ∩R = {0}. We show how the existence of such subspaces leads to various pathological properties of unbounded self-adjoint opera...

Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight. Using the Infeld-Hull factorization method, we generate from the raising and l...

For the hypoelliptic differential operators P = ∂ x + ( x∂y − x∂t 2 introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of k and l left open in the analysis, the operators P also fail to be analytic hypoelliptic (except for (k, l) = (0, 1)), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of e...

Abstract. We generalize earlier results of [2], [3], [6], [13], [14] to the case of functions of n-tuples of commuting self-adjoint operators. In particular, we prove that if a function f belongs to the Besov space B ∞,1(R ), then f is operator Lipschitz and we show that if f satisfies a Hölder condition of order α, then ‖f(A1 · · · , An)− f(B1, · · · , Bn)‖ ≤ constmax1≤j≤n ‖Aj −Bj‖ α for all n...

Interpolation inequalities of Gagliardo-Nirenberg type and compactness results for self-adjoint trace-class operators with finite kinetic energy are established. Applying these results to the minimization of various free energy functionals, we determine for instance stationary states of the Hartree problem with temperature corresponding to various statistics. Key-words. Compact self-adjoint ope...

Multidimensional transfer functions are an effective description of physical systems with time and space as independent variables. They allow to construct discrete models which are suitable for computer implementation. The derivation of multidimensional transfer functions is straightforward for systems with self-adjoint spatial operators. It is based on the application of suitable functional tr...

The property of cyclicity of a linear operator, or equivalently the property of simplicity of its spectrum, is an important spectral characteristic that appears in many problems of functional analysis and applications to mathematical physics. In this paper we study cyclicity in the context of rank-one perturbation problems for self-adjoint and unitary operators. We show that for a fixed non-zer...