We give an elementary proof of a formula recently obtained by Hammond, Moorhouse, and Robbins for the adjoint of a rationally induced composition operator on the Hardy space H2 [Christopher Hammond, Jennifer Moorhouse, and Marian E. Robbins, Adjoints of composition operators with rational symbol, J. Math. Anal. App., to appear]. We discuss some variants and implications of this formula, and use...

A self-adjoint operator A in a Krĕın space (K, [ · , · ]) is called partially fundamentally reducible if there exist a fundamental decomposition K = K+[+̇]K− (which does not reduce A) and densely defined symmetric operators S+ and S− in the Hilbert spaces (K+, [ · , · ]) and (K−,−[ · , · ]), respectively, such that each S+ and S− has defect numbers (1, 1) and the operator A is a self-adjoint ext...

The self-adjointness of an evolution operator corresponding to the model of the flat FRW universe with massless scalar field and cosmological constant quantized in the framework of loop quantum cosmology is studied in the case > 0. It is shown that, for < c 10:3‘ 2 Pl , the operator admits many self-adjoint extensions, each of the purely discrete spectrum. On the other hand for c the operator i...

Suppose L is a differential operator of order k. Moreover, let u be a function defined and k times continuously differentiable in the closure of a domain Ω of R. Provided the adjoint differential operator possesses a fundamental solution, we shall see that u can be recovered from Lu and the boundary values of u. Strictly speaking, we shall get an integral representation of u in form of the sum ...

Abstract In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace operator. The sufficient conditions for summability is obtained. For the orders of Riesz means, which greater than critical index N−1 2 we established the estimation for maximal operator of the Riesz means. Note that when order α of Riesz means is less than cr...

We discuss the Hamiltonian for a nonrelativistic electron with spin in the presence of a magnetic monopole and note that it is not self-adjoint in the lowest two angular momentum modes. We then use von Neumann's theory of self-adjoint extensions to construct a self-adjoint operator with the same functional form. In general, this operator will have eigenstates in which the lowest two angular mom...

In this notes unbounded regular operators on Hilbert C∗-modules over arbitrary C∗-algebras are discussed. A densely defined operator t possesses a densely defined adjoint operator if the graph of t is an orthogonal summand. Moreover, for a densely defined operator t the graph of t is orthogonally complemented if and only if t is regular. For a given C∗-algebra A any densely defined A-linear clo...

In this notes unbounded regular operators on Hilbert C∗-modules over arbitrary C∗-algebras are discussed. A densely defined closed operator t possesses a densely defined adjoint operator if the graph of t is an orthogonal summand. Moreover, for a densely defined closed operator t the graph of t is orthogonally complemented if and only if t is regular. For a given C*-algebra A any densely define...

In this tutorial we collect facts from the theory of self-adjoint operators, mostly with a view of what is relevant for applications in mathematical quantum mechanics, in particular for solving the Schrödinger equation. Specific topics include the spectral theorem and functional calculus for self-adjoint operators, Stone’s Theorem, Laplacians and Fourier transform, Duhamel’s formula and Dyson s...