Kazan (Volga region) Federal University, KFU
Form of presentationArticles in international journals and collections
Year of publication2019
  • Bikchentaev Ayrat Midkhatovich, author
  • Bibliographic description in the original language Airat M. Bikchentaev, On $\tau$-essentially invertibility of $\tau$-measurable operators // Internat. J. Theor. Phys. 2019. V. 58. No 12. https://doi.org/10.1007/s10773-019-04111-w
    Annotation Let ${\mathcal M}$ be a von Neumann algebra of operators on a Hilbert space and $\tau$ be a faithful normal semifinite trace on $\mathcal{M}$. Let $I$ be the unit of the algebra ${\mathcal M}$. A $\tau$-measurable operator $A$ is said to be {\it $\tau$-essentially right (or left) invertible} if there exists a $\tau$-measurable operator $B$ such that the operator $I-AB$ (or $I-BA$) is $\tau$-compact. A necessary and sufficient condition for an operator $A $ to be $\tau$-essentially left invertible is that $A^*A$ (or, equivalently, $\sqrt{A^*A}$) is $\tau$-essentially invertible. We present a sufficient condition that a $\tau$-measurable operator $A $ not be $\tau$-essentially left invertible. For $\tau$-measurable operators $A$ and $P=P^2$ the following conditions are equivalent: 1. $A$ is $\tau$-essential right inverse for $P$; 2. $A$ is $\tau$-essential left inverse for $P$; 3. $I-A, I-P $ are $\tau$-compact; 4. $PA$ is $\tau$-essential left inverse for $P$. For $\tau$-measurable operators $A=A^3$, $ B=B^3$ the following conditions are equivalent: 1. $B$ is $\tau$-essential right inverse for $A$; 2. $B$ is $\tau$-essential left inverse for~$A$. Pairs of faithful normal semifinite traces on $\mathcal{M}$ are considered.
    Keywords Hilbert space, von Neumann algebra, normal weight, semifinite trace, measure topology, $\tau$-measurable operator, $\tau$-compact operator, rearrangement, $\tau$-essentially invertible operator, idempotent.
    The name of the journal INT J THEOR PHYS
    Please use this ID to quote from or refer to the card https://repository.kpfu.ru/eng/?p_id=202091&p_lang=2
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