In this note, we give an example of a densely defined non-closable paranormal operator. Then, we give another example of a densely defined closable paranormal operator whose closure fails to be paranormal. These two examples are simpler than those which first appeared in [1]. It is worth noticing that our first example tells us that the adjoint of a densely defined paranormal operator may have a trivial domain.
In this note, we give an example of a densely defined closed one-to-one paranormal operator $T$ whose adjoint is also injective and paranormal, but $T$ fails to be normal.
In this paper, some sub-classes of paranormal weighted conditional expectation type operators, such as *-paranormal, quasi-*-paranormal and (n; k)-quasi-*-paranormal weighted conditional expectation type opera- tors on $L^2(Σ)$ are investigated. Also, some applications about the spectrum, point spectrum, joint point spectrum, approximate point spectrum and joint approximate point spectrum of these classes are presented.
In this article, we first prove the existence of an invariant subspace for a norm-attaining $\ast$-paranormal operator. Then give a representation for $\ast$-paranormal operators in the closure of absolutely norm-attaining operators and further study a few sufficient conditions for the normality of such operators. Finally, we discuss Toeplitz and Hankel $\ast$-paranormal operators in the closure of absolutely norm-attaining operators on the Hardy space.
For nonnegative integers $n$ and $k$, we introduce in this paper a new class of $(n,k)$-quasi-*-paranormal operators satisfying $$||T^{1+n}(T^{k}x)||^{1/(1+n)}||T^{k}x||^{n/(1+n)} \geq ||T^*(T^{k}x)|| \makebox{\ for all} x \in H.$$ This class includes the class of $n$-*-paranormal operators and the class of $(1,k)$-quasi-*-paranormal operators contains the class of $k$-quasi-*-class $A$ operators. We study basic properties of $(n,k)$-quasi-*-paranormal operators: (1) inclusion relations and examples; (2) a matrix representation; (3) joint (approximate) point spectrum and single valued extensio…
In this article, we give a representation of $\ast$-paranormal absolutely norm attaining operator. Explicitly saying, every $\ast$-paranormal absolutely norm attaining ($\mathcal{AN}$ in short) $T$ can be decomposed as $U\oplus D$, where $U$ is a direct sum of scalar multiple of unitary operators and $D$ is a $2\times 2$ upper diagonal operator matrix. By the representation it is clear that the class of $\ast$-paranormal $\mathcal{AN}$-operators is bigger than the class of normal $\mathcal{AN}$-operators but here we observe that a $\ast$-paranormal $\mathcal{AN}$-operator is normal if either i…
In this article, we show that a commuting pair $T=(T_1,T_2)$ of $\ast$-paranormal operators $T_1$ and $T_2$ with quasitriangular property satisfy the Weyl's theorem-I, that is $$σ_T(T)\setminusσ_{T_W}(T)=π_{00}(T)$$ and a commuting pair of paranormal operators satisfy Weyl's theorem-II, that is $$σ_T(T)\setminusω(T)=π_{00}(T),$$ where $σ_T(T),\, σ_{T_W}(T),\,ω(T)$ and $π_{00}(T)$ are the Taylor spectrum, the Taylor Weyl spectrum, the joint Weyl spectrum and the set consisting of isolated eigenvalues of $T$ with finite multiplicity, respectively. Moreover, we prove that Weyl's theorem-II holds …
In this article we discuss a few spectral properties of a paranormal closed operator (not necessarily bounded) defined in a Hilbert space. This class contains closed symmetric operators. First we show that the spectrum of such an operator is non empty. Next, we give a characterization of closed range operators in terms of the spectrum. Using these results we prove the Weyl's theorem: if $T$ is a densely defined closed, paranormal operator, then $σ(T)\setminusω(T)=π_{00}(T)$, where $σ(T), ω(T)$ and $π_{00}(T)$ denote the spectrum, Weyl spectrum and the set of all isolated eigenvalues with finit…
We provide several perturbation theorems regarding closable operators on a real or complex Hilbert space. In particular we extend some classical results due to Hess--Kato, Kato--Rellich and Wüst. Our approach involves ranges of matrix operators of the form $\left(\begin{array}{cc}\!\! I & A\!\!\\ \!\! -B& I\!\!\end{array}\right)$.
We present how to apply a Berberian's technique to asymmetric Putnam-Fuglede theorems. In particular, we proved that if $A, B \in B(H)$ belong to the union of classes of $*$-paranormal operators, p-hyponormal operators, dominant operators and operators of class Y and $AX = XB^*$ for some $X \in B(H)$, then $A^*X = XB$. Moreover, we gave a new counterexample for an asymmetric Putnam-Fuglede theorem for paranormal operators
Given a dissipative operator $A$ on a complex Hilbert space $\mathcal{H}$ such that the quadratic form $f\mapsto \mbox{Im}\langle f,Af\rangle$ is closable, we give a necessary and sufficient condition for an extension of $A$ to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.
In this paper, we are mainly concerned with studying arbitrary unbounded square roots of linear operators as well as some of their basic properties. The paper contains many examples and counterexamples. As an illustration, we give explicit everywhere defined unbounded non-closable $nth$ roots of the identity operator as well as the zero operator. We also show a non-closable unbounded operator without any non-closable square root. Among other consequences, we have a way of finding everywhere defined bijective operators, everywhere defined operators which are surjective without being injective a…
In a paper from 2016 D. R. Yafaev considers Hankel operators associated with Hamburger moment sequences q_n and claims that the corresponding Hankel form is closable if and only if the moment sequence tends to 0. The claim is not correct, since we prove closability for any indeterminate moment sequence but also for certain determinate moment sequences corresponding to measures with finite index of determinacy. It is also established that Yafaev's result holds if the moments satisfy \root{2n}\of{q_{2n}}=o(n).
Let (X,m) and (Y,n) be standard measure spaces. A function f in $L^\infty(X\times Y,m\times n)$ is called a (measurable) Schur multiplier if the map $S_f$, defined on the space of Hilbert-Schmidt operators from $L_2(X,m)$ to $L_2(Y,n)$ by multiplying their integral kernels by f, is bounded in the operator norm. The paper studies measurable functions f for which $S_f$ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of tw…
A linear relation, i.e., a multivalued operator $T$ from a Hilbert space ${\mathfrak H}$ to a Hilbert space ${\mathfrak K}$ has Lebesgue type decompositions $T=T_{1}+T_{2}$, where $T_{1}$ is a closable operator and $T_{2}$ is an operator or relation which is singular. There is one canonical decomposition, called the Lebesgue decomposition of $T$, whose closable part is characterized by its maximality among all closable parts in the sense of domination. All Lebesgue type decompositions are parametrized, which also leads to necessary and sufficient conditions for the uniqueness of such decomposi…
The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on $1$-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they may have positive two-dimensional Lebesgue measure and carry nontrivial $2$-forms. We prove that in this case the curl operator (and therefore also the exterior derivative on $1$-forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abst…
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.
In a paper from 2016 D. R. Yafaev initiated a study of closable Hankel forms associated with the moments $(m_n)$ of a positive measure with infinite support on the real line. If $m_n=o(1)$ Yafaev characterized the closure of the form based on earlier work on quasi-Carleman operators. We give a new proof of the description of the closure based entirely on moment considerations. The main purpose of the present paper is a description of the self-adjoint Hankel operators associated with closed Hankel forms in the Hilbert space of square summable sequences. We do this not only in the case $m_n=o(1)…
Aluthge transform of a bounded operator is generalized to the case of unbounded one. A formula for the Aluthge transform of a weighted shift on a directed tree is established and it is used to construct an example of a hyponormal operator whose Aluthge transform has trivial domain. It is proven that such an example can be also constructed in the class of composition operators. It is also shown that Aluthge transform of a closed, densely defined operator is not necessarily closable.
We find simple conditions for a non-negative Hankel quadratic form to be closable. Under some mild a priori assumption on the associated moments these sufficient conditions turn out to be also necessary. We also describe the domain of the corresponding closed form. This allows us to define unbounded non-negative Hankel operators under minimal assumptions on their matrix elements. The results obtained supplement the classical Widom condition for a Hankel operator to be bounded..
We show that a Krein-Feller operator is naturally associated to a fixed measure $μ$, assumed positive, $σ$-finite, and non-atomic. Dual pairs of operators are introduced, carried by the two Hilbert spaces, $L^{2}\left(μ\right)$ and $L^{2}\left(λ\right)$, where $λ$ denotes Lebesgue measure. An associated operator pair consists of two specific densely defined (unbounded) operators, each one contained in the adjoint of the other. This then yields a rigorous analysis of the corresponding $μ$-Krein-Feller operator as a closable quadratic form. As an application, for a given measure $μ$, including t…
We show that a semibounded Toeplitz quadratic form is closable in the space $\ell^2({\Bbb Z}_{+})$ if and only if its matrix elemens are Fourier coefficients of an absolutely continuous measure. We also describe the domain of the corresponding closed form. This allows us to define semibounded Toeplitz operators under minimal assumptions on their matrix elements.
Given a densely defined and closed operator $A$ acting on a complex Hilbert space $\mathcal{H}$, we establish a one-to-one correspondence between its closed extensions and subspaces $\mathfrak{M}\subset\mathcal{D}(A^*)$, that are closed with respect to the graph norm of $A^*$ and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of $A^*$. After this, we will express our results using the language of Gel'fand triples generalizing the well-known results for the selfadjoint case. As applications we construct: (i) a sequence o…
In this paper, we are concerned with conditions under which $[p(T)]^*=\bar{p}(T^*)$, where $p(z)$ is a one-variable complex polynomial, and $T$ is an unbounded, densely defined, and linear operator. Then, we deal with the validity of the identities $σ(AB)=σ(BA)$, where $A$ and $B$ are two unbounded operators. The equations $(TT^*)^*=TT^*$ and $(T^*T)^*=T^*T$, where $T$ is a densely defined closable operator, are also studied. A particular interest will be paid to the equation $T^*T=p(T)$ and its variants. Then, we have certain results concerning $nth$ roots of classes of normal and nonnormal (…
A linear operator $S$ in a complex Hilbert space $\hh$ for which the set $\dzn{S}$ of its $C^\infty$-vectors is dense in $\hh$ and $\{\|S^n f\|^2\}_{n=0}^\infty$ is a Stieltjes moment sequence for every $f \in \dzn{S}$ is said to generate Stieltjes moment sequences. It is shown that there exists a closed non-hyponormal operator $S$ which generates Stieltjes moment sequences. What is more, $\dzn{S}$ is a core of any power $S^n$ of $S$. This is established with the help of a weighted shift on a directed tree with one branching vertex. The main tool in the construction comes from the theory of in…
