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Register a new userGeneralized Polar Decompositions for Closed Operators in Hilbert Spaces and Some Applications
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We study generalized polar decompositions of densely defined, closed linear operators in Hilbert spaces and provide some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators.
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A bounded linear operator $ A$ on a Hilbert space $ mathcal H $ is said to be an $ EP $ (hypo-$ EP $) operator if ranges of $ A $ and $ A^* $ are equal (range of $ A $ is contained in range of $ A^* $) and $ A $ has a closed range. In this paper, we define $EP$ and hypo-$EP$ operators for densely defined closed linear operators on Hilbert spaces and extend results from bounded operator settings to (possibly unbounded) closed operator settings.
A notion of resolvent set for an operator acting in a rigged Hilbert space $D subset Hsubset D^times$ is proposed. This set depends on a family of intermediate locally convex spaces living between $D$ and $D^times$, called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.
In this paper, we investigate the perturbation for the Moore-Penrose inverse of closed operators on Hilbert spaces. By virtue of a new inner product defined on $H$, we give the expression of the Moore-Penrose inverse $bar{T}^dag$ and the upper bounds of $|bar{T}^dag|$ and $|bar{T}^dag -T^dag|$. These results obtained in this paper extend and improve many related results in this area.
We consider the decomposition of bounded linear operators on Hilbert spaces in terms of functions forming frames. Similar to the singular-value decomposition, the resulting frame decompositions encode information on the structure and ill-posedness of the problem and can be used as the basis for the design and implementation of efficient numerical solution methods. In contrast to the singular-value decomposition, the presented frame decompositions can be derived explicitly for a wide class of operators, in particular for those satisfying a certain stability condition. In order to show the usefulness of this approach, we consider different examples from the field of tomography.
Let $G$ be a locally compact abelian group with a Haar measure, and $Y$ be a measure space. Suppose that $H$ is a reproducing kernel Hilbert space of functions on $Gtimes Y$, such that $H$ is naturally embedded into $L^2(Gtimes Y)$ and is invariant under the translations associated with the elements of $G$. Under some additional technical assumptions, we study the W*-algebra $mathcal{V}$ of translation-invariant bounded linear operators acting on $H$. First, we decompose $mathcal{V}$ into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces $widehat{H}_xi$, $xiinwidehat{G}$, generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of $mathcal{V}$. Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to $mathcal{V}$, i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors.
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