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The purpose of this paper is to revise von Neumann’s characterizations of selfadjoint operators among symmetric ones. In fact, we do not assume that the underlying Hilbert space is complex, nor that the corresponding operator is densely defined, moreover, that it is closed. Following Arens, we employ algebraic arguments instead of the geometric approach of von Neumann using the Cayley transform.

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An extension of von Neumann’s characterization of essentially selfadjoint operators is given among not necessarily densely defined symmetric operators which are assumed to be closable. Our arguments are of algebraic nature and follow the idea of [1].

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Abstract.

We present some compactness properties of L-weakly and M-weakly compact operators on a Banach lattice under additional conditions. Thus, we can say that every bounded operator which commutes with any L-weakly or M-weakly compact operator have a non-trivial closed invariant subspace.

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Abstract

A nonnegative linear relation S in a Hilbert space ℌ is assumed to intertwine in a certain sense two bounded everywhere defined operators B and C. A related quotient of the range of S is then provided with a natural inner product and the operators B and C induce two operators on the completion space. This construction is used to show the existence of self-adjoint and nonnegative extensions of the linear relations B S and C S, respectively.

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Abstract

We give an extension of a classical result due to Krein on biorthogonal expansions of compact operators which are symmetrizable with respect to a nondegenerate positive operator. Our approach makes essential use of the spectral expansion of an appropriate compact selfadjoint operator, the existence of which is due to Dieudonné.

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Abstract

We characterize those positive (not necessarily densely defined) operators whose Krein–von Neumann extension, the smallest among all positive selfadjoint extensions, has closed range. In addition, we construct their Moore–Penrose pseudoinverse by employing factorization via an auxiliary Hilbert space. Other extremal extensions, in particular the Friedrichs extension, are also investigated from this point of view. As an application, new characterizations of essentially selfadjoint positive operators are presented.

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The notions of parallel sum and parallel difference of two nonnegative forms were introduced and studied by Hassi, Sebestyén, and de Snoo in [13] and [14]. In this paper we consider the parallel subtraction with much circumstances. Criteria are established for the solvability of the equation with an unknown when and are given. We identify as the minimal solution, and characterize all the solutions under the assumption where λ>1. The Galois correspondence induced by the map is also studied. We show that if the equation is solvable, then there is a unique -closed solution, namely . Finally, we consider some extremal problems such as the extreme points of the interval , and the characterization of the minimal forms in terms of the parallel sum.

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