On the mappings connected with parallel addition of nonnegative\n operators

2015/10/05 by Yu. M. Arlinskiĭ, Arlinskiĭ, Yu. M.
Computer Science · Mathematics · #46B25 #47A05 #47A64 #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1510.01282

openalex publication_date 2015/10/05 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We study a mapping \τ_G of the cone mathbf B⁺( mathcal H) of\nbounded nonnegative self-adjoint operators in a complex Hilbert space\n mathcal H into itself. This mapping is defined as a strong limit of\niterates of the mapping mathbf B⁺( mathcal H) ni\nX\↦\μ_G(X)=X-X:G\∈ mathbf B⁺( mathcal H), where G\∈ mathbf\nB⁺( mathcal H) and X:G is the parallel sum. We find explicit expressions\nfor \τ_G and establish its properties. In particular, it is shown that\n\τ_G is sub-additive, homogeneous of degree one, and its image coincides\nwith set of its fixed points which is the subset of mathbf B⁺( mathcal\nH), consisting of all Y such that rm ran , Y^1/2\∩ rm ran ,\nG^1/2= 0 . Relationships between \τ_G and Lebesgue type decomposition\nof nonnegative self-adjoint operator are established and applications to the\nproperties of unbounded self-adjoint operators with trivial intersections of\ntheir domains are given.\n

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