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...

We provide, by a resolvent Krĕın-like formula, all selfadjoint extensions of the symmetric operator S obtained by restricting the self-adjoint operator A : D(A) ⊆ H → H to the dense, closed with respect to the graph norm, subspace N ⊂ D(A). Neither the knowledge of S∗ nor of the deficiency spaces of S is required. Typically A is a differential operator and N is the kernel of some trace (restric...

Spectral problem for a self-adjoint third-order differential operator with non-local potential on finite interval is studied. Elementary functions that are analogues of sines and cosines such operators described. Direct inverse problems solved.

The spectral problem (A + V (z))ψ = zψ is considered with A, a self-adjoint operator. The perturbation V (z) is assumed to depend on the spectral parameter z as resolvent of another self-adjoint operator A ′ : V (z) = −B(A ′ − z) −1 B *. It is supposed that the operator B has a finite Hilbert-Schmidt norm and spectra of the operators A and A ′ are separated. Conditions are formulated when the p...

The perturbation theory of linear operators concerns itself with the eigenvalues and eigenelements of a variable operator. If T(e) is an operator depending on a real parameter e, the relation between the eigenvalues and eigenelements of the perturbed operator T(e) (e?^0) and those of the unperturbed operator 7\0) is of particular interest. This theory finds considerable application in the field...

In this paper has been studied the wave equation in some non-classic cases. In the  rst case boundary conditions are non-local and non-periodic. At that case the associated spectral problem is a self-adjoint problem and consequently the eigenvalues are real. But the second case the associated spectral problem is non-self-adjoint and consequently the eigenvalues are complex numbers,in which two ...

Given two self-adjoint, positive, compact operators A,B on a separable Hilbert space, we show that there exists a self-adjoint, positive, compact operator C commuting with B such that limt→∞ ||(e Bt 2 ee Bt 2 ) 1 t − e || = 0. 1

Let V be a finite-dimensional vector space, either real or complex, and equipped with an inner product 〈· , ·〉. Let A : V → V be a linear operator. Recall that the adjoint of A is the linear operator A : V → V characterized by 〈Av, w〉 = 〈v, Aw〉 ∀v, w ∈ V (0.1) A is called self-adjoint (or Hermitian) when A = A. Spectral Theorem. If A is self-adjoint then there is an orthonormal basis (o.n.b.) o...

We consider a positive self-adjoint operator A and formal rank one pertubrations

Maximal and atomic Hardy spaces Hp and H A, 0 < p ≤ 1, are considered in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. It is shown that Hp = H A with equivalent norms.