Some representations of moore-penrose inverse for the sum of two operators and the extension of the fill-fishkind formula

Abdessalam Kara; Said Guedjiba

doi:10.3934/naco.2021015

Article Contents

2022, Volume 12, Issue 3: 469-480. Doi: 10.3934/naco.2021015

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Some representations of moore-penrose inverse for the sum of two operators and the extension of the fill-fishkind formula

Abdessalam Kara and
Said Guedjiba

University of Batna 2, Faculty of Mathematics and Computer Sciences, Department of Mathematics, Algeria

Received: April 30, 2020

Revised: March 31, 2021

Early access: May 2021

Published: September 2022

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Abstract

In the setting of arbitrary Hilbert spaces, we give a representation of M-P inverse of the sum of linear operators $ A+B $ under suitable conditions. Based on the full-rank decomposition of an operator, we prove that the extension of the Fill-Fishkind formula for $ A $ and $ B $ with closed ranges, remains valid, keeping the same conditions of Fill-Fishkind formula for two matrices, also we obtain an analogous formula under the Fill-Fishkind conditions, beyond we derive some representations of M-P inverse of a 2-by-2 block operator with disjoint ranges.

Keywords:

Moore-Penrose inverse; Closed range operator. Sum of operators,
Disjoint ranges.

Mathematics Subject Classification: Primary: 15A09, Secondary: 47A05.

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