Related papers: On The Absolute Value of Unbounded Operators

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 this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of…

In this paper, we characterize absolute norm-attainability for compact hyponormal operators. We give necessary and sufficient conditions for a bounded linear compact hyponormal operator on an infinite dimensional complex Hilbert space to be…

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…

Let $x=a+ib$ be a complex number, so we have the following inequality $$(1/\sqrt{2})|a+b|\leq |x|\leq |a|+|b|$$ We give an operator version of above inequality. Also we obtain some results for normal operators.

Assume that $Au=f,\quad (1)$ is a solvable linear equation in a Hilbert space $H$, $A$ is a linear, closed, densely defined, unbounded operator in $H$, which is not boundedly invertible, so problem (1) is ill-posed. It is proved that the…

In this paper, we further investigate the problem of commutativity up to a factor (or $\lambda$-commutativity) in the setting of bounded and unbounded linear operators in a complex Hilbert space. The results are based on a new approach to…

We generalize Moore's nonstandard proof of the Spectral theorem for bounded self-adjoint operators to the case of unbounded operators. The key step is to use a definition of the nonstandard hull of an internally bounded self-adjoint…

This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a…

We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove…

The paper introduces unbounded antilinear operators on Hilbert spaces and develops their fundamental theory. In particular, we establish a closed range theorem, a polar decomposition theorem, and the convexity of the numerical range for…

In this paper, we establish several characterizations of the $A$-parallelism of bounded linear operators with respect to the seminorm induced by a positive operator $A$ acting on a complex Hilbert space. Among other things, we investigate…

This paper has a twofold purpose: to present an overview of the theory of absolutely summing operators and its different generalizations for the multilinear setting, and to sketch the beginning of a research project related to an objective…

Let $X$ be a vector lattice and $(E,\tau)$ be a locally solid vector lattice. An operator $T:X\to E$ is said to be $ob$-bounded if, for each order bounded set $B$ in $X$, $T(B)$ is topologically bounded in $E$. In this paper, we study on…

We prove upper and lower bounds for sums of eigenvalues of Lieb-Thirring type for non-self-adjoint Schr\"odinger operators on the half-line. The upper bounds are established for general classes of integrable potentials and are shown to be…

In this paper, we show a new and classic version of the celebrated Fuglede Theorem in an unbounded setting. A related counterexample is equally presented. In the second strand of the paper, we give a pair of a closed and self-adjoint…

In the present paper the unconditional convergence and the invertibility of multipliers is investigated. Multipliers are operators created by (frame-like) analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or…

We study convergence of operator families of the form $A_\beta = A + \beta B$ towards an effective operator defined on $\ker(B)$, as the coupling constant $\beta$ tends to infinity. Crucially, we focus on the setting where neither $A$ nor…

We characterize the space $BV(I)$ of functions of bounded variation on an arbitrary interval $I\subset \mathbb{R}$, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator $M_R$ from $BV(I)$ into the…

In this article, we characterize absolutely norm attaining normal operators in terms of the essential spectrum. Later we prove a structure theorem for hyponormal absolutely norm attaining (or $\mathcal{AN}$-operators in short) and deduce…