Two simple methods for approximate determination of eigenvalues and eigenvectors of linear self-adjoint operators are considered in the following two cases: (i) lower-upper bound ${\lambda }_1$ of the spectrum $\sigma \left(A\right)$ of $A$ is an isolated point of $\sigma \left(A\right)$; (ii) ${\lambda }_1$ (not necessarily an isolated point of $\sigma \left(A\right)$ with finite multiplicity) is an eigenvalue of $A$.

The Cauchy dual operator T’, given by $T{(T*T)}^{-1}$, provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T’ admits the Wold-type decomposition (see Definitions 1.1 and 1.2 below)....

Per funzioni opportune $f,g$ si ottiene una formula di Parseval ${⟨{𝐑}^Q,𝒬f𝒬g⟩}_{\lambda }=⟨{\mathrm{\Delta }}_Q^{-1/2}f,{\mathrm{\Delta }}_Q^{-1/2}g⟩$ per operatori differenziali singolari di tipo dell'operatore radiale di Laplace-Beltrami. ${𝐑}^Q$ è una funzione spettrale generalizzata di tipo Marčenko e può essere rappresentata per mezzo di un certo nucleo della trasmutazione.

Let $P$ be a selfadjoint classical pseudo-differential operator of order $\>1$ with non-negative principal symbol on a compact manifold. We assume that $P$ is hypoelliptic with loss of one derivative and semibounded from below. Then exp$(-tP)$, $t\ge 0$, is constructed as a non-classical Fourier integral operator and the main contribution to the asymptotic distribution of eigenvalues of $P$ is computed. This paper is a continuation of a series of joint works with A. Menikoff.

A new proof of the monotonicity of the Temple quotients for the computation of the dominant eigenvalue of a bounded linear normal operator in a Hilbert space is given. Another goal of the paper is a precise analysis of the length of the interval for admissible shifts for the Temple quotients.