We provide necessary and sufficient conditions for a pair $S,T$ of Hilbert space operators in order that they satisfy $S^*=T$ and $T^*=S$. As a main result we establish an improvement of von Neumann's classical theorem on the positive self-adjointness of $S^*S$ for two variables. We also give some new characterizations of self-adjointness and skew-adjointness of operators, not requiring their symmetry or skew-symmetry, respectively.
We consider positive semidefinite kernels valued in the $*$-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of $*$-semigroups. A rather general dilation theorem is stated and proved: for these kind of kernels, representations of the $*$-semigroup on either the VE-spaces of linearisation of the kernels or on their reproducing kernel VE-spaces are obtainable. We point out the reproducing kernel fabric of dilation theory and we show that the general theorem unifies many dilation results at the non topological level.
If a Banach-space operator has a complemented range, then its normed-space adjoint has a complemented kernel and the converse holds on a reflexive Banach space. It is also shown when complemented kernel for an operator is equivalent to complemented range for its normed-space adjoint. This is applied to compact operators and to compact perturbations. In particular, compact perturbations of semi-Fredholm operators have complemented range and kernel for both the perturbed operator and its normed-space adjoint
In this paper we study truncated moment problems for $J$-self-adjoint, $J$-skew-self-adjoint and $J$-unitary operators. Conditions of the solvability are given. Some canonical solutions of the moment problems are constructed. As a by-product, some extension results for $J$-skew-symmetric and $J$-isometric operators are obtained.
Closed operators in Hilbert space defined by a non-self-adjoint resolution of the identity $\{X(λ)\}_{λ\in {\mb R}}$, whose adjoints constitute also a resolution of the identity, are studied . In particular, it is shown that a closed operator $B$ has a spectral representation analogous to the familiar one for self-adjoint operators if and only if $B=TAT^{-1}$ where $A$ is self-adjoint and $T$ is a bounded operator with bounded inverse.
We consider the formal prolate spheroid differential operator on a finite symmetric interval and describe all its self-adjoint boundary conditions. Only one of these boundary conditions corresponds to a self-adjoint differential operator which commute with the Fourier operator truncated on the considered finite symmetric interval.
In this paper we study the self-adjoint and co-isometry composition and weighted composition operator on $H_E(ζ).$ We also discuss the conditions under which adjoint operator of a composition and weighted composition operator on $H_E(ζ)$ to be some composition and weighted composition operator, respectively.
The paper is devoted to a development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of C-symmetries. The main attention is paid to the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The general results are specialized further by studying in detail the case where S has defect numbers <2,2>.
In this article, the self-adjoint extensions of symmetric operators satisfying anti-commutation relations are considered. It is proven that an anti-commutative type of the Glimm-Jaffe-Nelson commutator theorem follows. Its application to an abstract Dirac operator is also considered.
Given the symmetric operator $A_N$ obtained by restricting the self-adjoint operator $A$ to $N$, a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint $A_N^*$ and the corresponding Weyl function. These objects provide us with the self-adjoint extensions of $A_N$ and their resolvents.
In this paper we obtain characterizations for adjoint of a composition and weighted composition operator to be composition and weighted composition operator on $F_ψ^2,$ respectively. We study the co-isometry composition and weighted composition operators on $F_ψ^2.$
In this paper we consider the order-like relation for self-adjoint operators on some Hilbert space. This relation is defined by using Jensen inequality. We will show that under some assumptions this relation is antisymmetric.
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 weighted composition operators. Normality of these operators is also investigated.
In a real Hilbert spaces H a smooth operator F is studied, whose derivative at each point of its domain is a symmetric operator. In terms of abstract boundary conditions locally self-adjoint extensions of this operator are described. We use some concepts and facts from symplectic differential geometry.
We get a criterion for 0 to be in the essential spectrum of a sum of self-adjoint operators whose pairwise products are compact. Using this result, we obtain necessary and sufficient conditions for the sum of ranges of such operators to be closed.
In this article, we prove and disprove several generalizations of unbounded versions of the Fuglede-Putnam theorem. As applications, we give conditions guaranteeing the commutativity of a bounded self-adjoint operator with an unbounded closed symmetric operator.
We generalize the classical notion of adjoint of a linear operator and the Aron-Schottenloher notion of adjoint of a homogeneous polynomial. The general notion is shown to enjoy several properties enjoyed by the classical ones, nevertheless differences between the two theories are detected. The proofs of both positive and negative results are not simple adaptations of the linear cases, actually nonlinear arguments are often required. Applications of the generalized adjoints to Lindström-Schlüchtermann type theorems for composition operators are provided.
We are interested in an open question raised by Fong-Tsui (dating back to the beginning of the eighties of last century) as to whether a bounded operator whose absolute value is less than the absolute value of its real part is self-adjoint. The analogue in the unbounded operators setting is also treated.
In this note, we mainly investigate the validity of the identities $(TT^*)^*=TT^*$ and $(T^*T)^*=T^*T$, where $T$ is a densely defined closable (or symmetric) operator.
Building on techniques developed by Cowen and Gallardo-Gutiérrez, we find a concrete formula for the adjoint of a composition operator with rational symbol acting on the Hardy space $H^{2}$. We consider some specific examples, comparing our formula with several results that were previously known.
Maximal and atomic Hardy spaces Hp and HAp , 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 = HAp with equivalent norms.
A new geometric proof of the spectral theorem for unbounded self-adjoint operators A in a Hilbert space H is given based on a splitting of A in positive and negative parts A+ and A-. For both operators A+ and A- the spectral family can be defined immediately and then put together to become the spectral family of A.
Given a finite set $X\subseteq\R$ we characterize the diagonals of self-adjoint operators with spectrum $X$. Our result extends the Schur-Horn theorem from a finite dimensional setting to an infinite dimensional Hilbert space analogous to Kadison's theorem for orthogonal projections and the second author's result for operators with three point spectrum.
Let $A$ be a $ν$-vector of self-adjoint, pairwise commuting operators and $B$ a bounded operator of class $C^{n_0}(A)$. We prove a Taylor-like expansion of the commutator $[B,f(A)]$ for a large class of functions $f\colon\mathbm{R}^ν\to \mathbm{R}$, generalising the one-dimensional result where $A$ is just a self-adjoint operator. This is done using almost analytic extensions and the higher-dimensional Helffer-Sjöstrand formula.
Symplectic self-adjointness of Hamiltonian operator matrices is studied, which arises in symplectic elasticity and optimal control. For the cases of diagonal domain and off-diagonal domain, necessary and sufficient conditions are shown. The proofs use Frobenius-Schur fractorizations of unbounded operator matrices. Under additional assumptions, sufficient conditions based on perturbation method are obtained. The theory is applied to a problem in symplectic elasticity.
An adjoint pair is a pair of densely defined linear operators $A, B$ on a Hilbert space such that $\langle Ax,y\rangle=\langle x,By\rangle$ for $x\in \cD(A), y \in \cD(B).$ We consider adjoint pairs for which $0$ is a regular point for both operators and associate a boundary triplet to such an adjoint pair. Proper extensions of the operator $B$ are in one-to-one correspondence $T_\cC\leftrightarrow \cC $ to closed subspaces $\cC$ of $\cN(A^*)\oplus\cN(B^*)$. In the case when $B$ is formally normal and $\cD(A)=\cD(B)$, the normal operators $T_\cC$ are characterized. Next we assume that $B$ has …
This is a continuation of our previous paper. We consider a certain order-like relation for positive operators on a Hilbert space. This relation is defined by using the Jensen inequality with respect to the square-root function. We show that this relation is antisymmetric if the operators are invertible.
In this paper, we establish results about operators similar to their adjoints. This is carried out in the setting of bounded and also unbounded operators on a Hilbert space. Among the results, we prove that an unbounded closed operator similar to its adjoint, via a cramped unitary operator, is self-adjoint. The proof of this result works also as a new proof of the celebrated result by Berberian on the same problem in the bounded case. Other results on similarity of hyponormal unbounded operators and their self-adjointness are also given, generalizing famous results by Sheth and Williams.
Let $B$ be a bounded self-adjoint operator and let $A$ be a nonnegative self-adjoint unbounded operator. It is shown that if $BA$ is normal, it must be self-adjoint and so must be $AB$. Commutativity is necessary and sufficient for this result. If $AB$ is normal, it must be self-adjoint and $BA$ is essentially self-adjoint. Although the two problems seem to be alike, two different and quite interesting approaches are used to tackle each one of them.
A theorem that is of aid in computing the domain of the adjoint operator is provided. It may serve e.g. as a criterion for selfadjointness of a symmetric operator, for normality of a formally normal operator or for $H$--selfadjointness of an $H$--symmetric operator. Differential operators and operators given by an infinite matrix are considered as examples.
In this paper we {\em discuss} diverse aspects of mutual relationship between adjoints and formal adjoints of unbounded operators bearing a matrix structure. We emphasize on the behaviour of row and column operators as they turn out to be the germs of an arbitrary matrix operator, providing most of the information about the latter {as it is the troublemaker}.
Devinatz, Nussbaum and von Neumann established some important results on the strong commutativity of self-adjoint and normal unbounded operators. In this paper, we prove results in the same spirit.
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 H^2. We discuss some variants and implications of this formula, and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator.
Let $u=\int_{-\infty}^{+\infty}λdE_λ$ be a self-adjoint operator in a Hilbert space $H$. Our purpose is to provide a non-standard description of the spectral family $(E_λ)$ and the generalized Gelfand eigenvectors.
We construct operators which factorize the transfer function associated with a non-self-adjoint 2x2 operator matrix whose diagonal entries can have overlapping spectra and whose off-diagonal entries are unbounded operators.
In this paper, we mainly deal with sequences of bounded linear operators on Hilbert space. The main result is the so-called squeeze theorem (or sandwich rule) for convergent sequences of self-adjoint operators. We show that this theorem remains valid for all three main topologies on $B(H)$.
We extend Krupnik's criterion of self-adjointness of the Cauchy singular integral operator to the case of finitely connected domains. The main aim of the paper is to present a new approach for proof of the criterion.
We study finitely cyclic self-adjoint operators in a Hilbert space, i.e. self-adjoint operators that posses such a finite subset in the domain that the orbits of all its elements with respect to the operator are linearly dense in the space. One of the main goals here is to obtain the representation theorem for such operators in a form analogous to the one well-known in the cyclic self-adjoint operators case. To do this, we present here a detailed introduction to matrix measures, to the matrix measure $L^2$ spaces, and to the multiplication by scalar functions operators in such spaces. This all…
In this expository paper we answer two fundamental questions concerning discrete magnetic Schrödinger operator associated with weighted graphs. We discuss when formal expressions of such operators give rise to self-adjoint operators, i.e., when they have self-adjoint restrictions. If such self-adjoint restrictions exist, we explore when they are unique.
This paper is devoted to self-adjoint cyclically compact operators on Hilbert--Kaplansky module over a ring of bounded measurable functions. The spectral theorem for such a class of operators are given. We apply this result to partial integral equations on the space with mixed norm of measurable functions and to compact operators relative to von Neumann algebras. We will give a condition of solvability of partial integral equations with self-adjoint kernel. Moreover, a general form of compact operators relative to a type I von Neumann algebra is given.
Let $H_0$ and $H$ be self-adjoint operators in a Hilbert space. In the scattering theory framework, we describe the essential spectrum of the difference $φ(H)-φ(H_0)$ for piecewise continuous functions $φ$. This description involves the scattering matrix for the pair $H$, $H_0$.
In this paper we give and prove a criterion for the normality of unbounded closed operators, which is a sort of a maximality result which will be called "double maximality". As applications, we show, under some assumptions, that the sum of two symmetric operators is essentially self-adjoint; and that the sum of two unbounded normal operators is essentially normal. Some other important results are also established.
Spectral theory and functional calculus for unbounded self-adjoint operators on a Hilbert space are usually treated through von Neumann's Cayley transform. Based on ideas of Woronowicz, we redevelop this theory from the point of view of multiplier algebras and the so-called bounded transform (which establishes a bijective correspondence between closed operators and pure contractions). This also leads to a simple account of the affiliation relation between von Neumann algebras and self-adjoint operators.
We consider the self-adjoint Dirac operators on a finite interval with summable matrix-valued potentials and general boundary conditions. For such operators, we study the inverse problem of reconstructing the potential and the boundary conditions of the operator from its eigenvalues and suitably defined norming matrices.
We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we obtain the Lieb--Thirring inequalities for such operators. The spectral enclosure for the discrete spectrum and embedded eigenvalues are also discussed.
The work is devoted to the study of the similarity of a correct restriction to some self-adjoint operator in the case when the minimal operator is symmetric. The resulting theorem was applied to the Sturm-Liouville operator and the Laplace operator. It is shown that the spectrum of a non self-adjoint singularly perturbed operator is real and the corresponding system of eigenvectors forms a Riesz basis.
We characterize the convex-cyclic weighted composition operators $W_{(u,ψ)}$ and their adjoints on the Fock space in terms of the derivative powers of $ ψ$ and the location of the eigenvalues of the operators on the complex plane. Such a description is also equivalent to identifying the operators or their adjoints for which their invariant closed convex sets are all invariant subspaces. We further show that the space supports no supercyclic weighted composition operators with respect to the pointwise convergence topology and hence with the weak and strong topologies and answers a question rais…
