Unbounded Normal Operators Research Articles

Generalizations of Weiss’ version of the Fuglede–Putnam theorem for unbounded operators

The present paper discusses Weiss’ version of the Fuglede–Putnam theorem. We establish three generalizations of this version, each of which involves unbounded normal operators. The first generalization expands upon Weiss’ theorem by incorporating a single unbounded normal operator, while the second involves normal powers. The third generalization deals with two unbounded normal operators. These generalizations have several applications and are accompanied by counterexamples.

Spectral representation of absolutely minimum attaining unbounded normal operators

Spectral representation of absolutely minimum attaining unbounded normal operators

Adjoint pairs and unbounded normal operators

An adjoint pair is a pair of densely defined closed linear operatorsA,B on a Hilbert space such that langle Ax,yrangle = langle x,Byrangle for x in mathcal{D} (A),y in mathcal{D} (B) . We consider adjoint pairs for which 0 is a regular point for bothoperators and associate a boundary triplet to such an adjoint pair. Properextensions of the operator B are in one-to-one correspondence Tcleftrightarrow mathcal{C} to closed subspaces mathcal{C} of mathcal{N}(A^{*}) oplusmathcal{N}(B^{*}). In the case when B is formallynormal andmathcal{D}(A) =mathcal{D}(B) , the normal operators Tc are characterized.Next we assume that B has an extension to a normal operator withbounded inverse. Then the normal operators Tc are described and thecase when mathcal{N}(A^{*}) has dimension one is treated.

The spectral theorem for normal operators on a Clifford module

In this paper, using the recently discovered notion of the S-spectrum, we prove the spectral theorem for a bounded or unbounded normal operator on a Clifford module (i.e., a two-sided Hilbert module over a Clifford algebra based on units that all square to be \(-1\)). Moreover, we establish the existence of a Borel functional calculus for bounded or unbounded normal operators on a Clifford module. Towards this end, we have developed many results on functional analysis, operator theory, integration theory and measure theory in a Clifford setting which may be of an independent interest. Our spectral theory is the natural spectral theory for the Dirac operator on manifolds in the non-self adjoint case. Moreover, our results provide a new notion of spectral theory and a Borel functional calculus for a class of n-tuples of commuting or non-commuting operators on a real or complex Hilbert space. Moreover, for a special class of n-tuples of operators on a Hilbert space our results provide a complementary functional calculus to the functional calculus of J. L. Taylor.

DECOMPOSITION OF NORMAL OPERATORS AND ITS APPLICATION TO SPECTRAL THEOREM

A decomposition theorem for self-adjoint operators proved by Riesz and Lorch is extended to normal operators. This extension gives a new proof of the spectral theorem for unbounded normal operators.

On the absolute value of unbounded operators

The primary purpose of the present paper is to investigate when relations of the types |AB|=|A||B|, |A±B|⩽|A|+|B|, ||A|−|B||⩽|A±B| and |¯¯¯¯¯¯¯¯¯¯¯ReA|⩽|A| (among others) hold in an unbounded operator setting. As consequences, we obtain a characterization of (unbounded) self-adjointness as well as a characterization of invertibility for the class of unbounded normal operators. Some examples accompany our results.

Spectral representations of normal operators in quaternionic Hilbert spaces via intertwining quaternionic PVMs

The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann’s foundational works in the thirties. The absence of a suitable quaternionic version of spectrum prevented the full development of the theory. The first rigorous quaternionic formulation has started only in 2007 with the definition of the spherical spectrum of a quaternionic operator based on a quadratic version of resolvent operator. The relevance of this notion is proved by the existence of a quaternionic continuous functional calculus and a theory of quaternionic semigroups relying upon it. A problem of the quaternionic formulation is the description of composite quantum systems in the absence of a natural tensor product due to non-commutativity of quaternions. A promising tool towards a solution is a quaternionic projection-valued measure (PVM), making possible a tensor product of quaternionic operators with physical relevance. A notion with this property, called intertwining quaternionic PVM, is presented here. This foundational paper aims to investigate the interplay of this new mathematical object and the spherical spectral features of quaternionic generally unbounded normal operators. We discover, in particular, the existence of other spectral notions equivalent to the spherical ones, but based on a standard non-quadratic notion of resolvent operator.

The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum

In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The S-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators. The lack of a suitable notion of spectrum was a major obstruction to fully understand the spectral theorem for quaternionic normal operators. A prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.

A Generalization of the Fuglede-Putnam Theorem to Unbounded Operators

Let N,M be unbounded normal operators in a Hilbert space and let T be a closed operator whose domain D(T) contains the domain of N, and the domain D(T*) contains the domain of M. It is shown that if TN⊆MT, then TN*⊆M*T.

A criterion for the normality of unbounded operators and applications to self-adjointness

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.

Commutativity of unbounded normal and self-adjoint operators and applications

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.

On a Riesz basis of finite-dimensional invariant subspaces and application to Gribov operator in Bargmann space

In this paper, we are mainly concerned with a new class of unbounded perturbations of unbounded normal operators. We give a description of the changed spectrum and we establish different conditions in terms of the spectrum to prove the existence of Riesz basis of finite-dimensional invariant subspaces of generalized eigenvectors. The obtained results are of importance for application to a non-self-adjoint Gribov operator in Bargmann space.

Unbounded products of operators and connections to Dirac-type operators

Let A and B be two densely defined unbounded closeable operators in a Hilbert space such that their unbounded operator products AB and BA are also densely defined. Then all four operators possess adjoints and we obtain new inclusion bounds for the operator product closures A¯B¯ and AB¯ in terms of new relations among the operator adjoints. These in turn lead to sharpened understandings for when products of unbounded self-adjoint and unbounded normal operators are self-adjoint and normal. They also clarify certain operator-product issues for Dirac operators.

Some Remarks on the Spectral Theory and Commutativity of Unbounded Normal Operators

In this paper we give a development of the spectral theory and commutativity properties of unbounded normal operators in the style of Kadison and Ringrose.

Quaternionic Cayley transform revisited

A Cayley type transform in the context of quaternions (previously introduced by the author) is revisited. A simplified definition is given and shorter proofs are exhibited. As new results, the range of this transform restricted to a class of unbounded normal operators is described, and an application to a moment problem with constraints is given. The extension problem to normal operators in this new context is also revisited.

Unbounded Normal Operators in Octonion Hilbert Spaces and Their Spectra

Affiliated and normal operators in octonion Hilbert spaces are studied. Theorems about their properties and of related algebras are demonstrated. Spectra of unbounded normal operators are investigated.

A note on the Fuglede–Putnam theorem

We prove the following generalization of the Fuglede–Puntam theorem. Let N be an unbounded normal operator in the Hilbert space, and let A be an unbounded self-adjoint operator such that \(D(N)\subseteq D(A)\). Then, \( AN\subseteq N^*A \Rightarrow AN^*\subseteq NA.\)

Perturbation theory for normal operators

Let $E \ni x\mapsto A(x)$ be a $\mathscr {C}$-mapping with its values being unbounded normal operators with common domain of definition and compact resolvent. Here $\mathscr {C}$ stands for $C^\infty$, $C^\omega$ (real analytic), $C^{[M]}$ (Denjoy–Carleman of Beurling or Roumieu type), $C^{0,1}$ (locally Lipschitz), or $C^{k,\alpha }$. The parameter domain $E$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We completely describe the $\mathscr {C}$-dependence on $x$ of the eigenvalues and the eigenvectors of $A(x)$. Thereby we extend previously known results for self-adjoint operators to normal operators, partly improve them, and show that they are best possible. For normal matrices $A(x)$ we obtain partly stronger results.

On commuting operator exponentials, II

We prove that, under sufficient conditions on the spectra, $$ e^Me^N\subseteq e^Ne^M\Rightarrow MN\subseteq NM , $$ where N is an unbounded normal operator and M is a bounded normal operator in the Hilbert space.

Spectral approximations of unbounded nonselfadjoint operators

We consider the operator \(A=S+B,\) where \(S\) is an unbounded normal operator in a separable Hilbert space \(H,\) having a compact inverse one and \(B\) is a linear operator in \(H,\) such that \(BS^{-1} \) is compact. Let \(\{e_k\}_{k=1}^\infty \) be the normalized eigenvectors of \(S\) and \(B\) be represented in \(\{e_k\}_{k=1}^\infty \) by a matrix \((b_{jk})_{j,k=1}^\infty .\) We approximate the eigenvalues of \(A\) by a combination of the eigenvalues of \(S\) and the eigenvalues of the finite matrix \({(b_{jk})}_{j,k=1}^{n}.\) Applications of to differential operators are also discussed.