On Rank One H−3-Perturbations of Positive Self–adjoint Operators

A note on $lambda$-Aluthge transforms of operators

Let $A=U|A|$ be the polar decomposition of an operator $A$ on a Hilbert space $mathscr{H}$ and $lambdain(0,1)$. The $lambda$-Aluthge transform of $A$ is defined by $tilde{A}_lambda:=|A|^lambda U|A|^{1-lambda}$. In this paper we show that emph{i}) when $mathscr{N}(|A|)=0$, $A$ is self-adjoint if and only if so is $tilde{A}_lambda$ for some $lambdaneq{1over2}$. Also $A$ is self adjoint if and onl...

Rank one perturbations of self-adjoint operators

Canonical Systems and Finite Rank Perturbations of Spectra

We use Rokhlin’s Theorem on the uniqueness of canonical systems to find a new way to establish connections between Function Theory in the unit disk and rank one perturbations of self-adjoint or unitary operators. In the n-dimensional case, we prove that for any cyclic self-adjoint operator A, operator Aλ = A + Σ k=1 λk(·, φk)φk is pure point for a.e. λ = (λ1, λ2, ..., λn) ∈ R n iff operator Aη ...

A Krein-like Formula for Singular Perturbations of Self-adjoint Operators and Applications

Given a self-adjoint operator A : D(A) ⊆ H → H and a continuous linear operator τ : D(A) → X with Range τ ′ ∩ H = {0}, X a Banach space, we explicitly construct a family A Θ of self-adjoint operators such that any A Θ coincides with the original A on the kernel of τ . Such a family is obtained by giving a Krĕın-like formula where the role of the deficiency spaces is played by the dual pair (X ,...

Singularly Perturbed Self-Adjoint Operators in Scales of Hilbert spaces

Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of A by the same formula. As an application the one-dimensional Schrödinger operator with generalized zero-range potential is consider...