ON REPRESENTATION OF ONE CLASS OF SCHMIDT OPERATORS
Orazov I. Shaldanbaeva A.A.
30 September 2021al-Farabi Kazakh State National University
KazNU Bulletin. Mathematics, Mechanics, Computer Science Series
2021#111Issue 352 - 64 pp.
In this paper, unitary symmetrizers are considered. It is well known that using Newton operator algorithm, similar to the usual Newton algorithm, for extracting the square root, one can prove that for every Hermitian operator T ≥ 0, there exists a unique Hermitian operator S ≥ 0 such that T = S2. Moreover, S commutes with every bounded operator R with which commutes T. The operator S is called a square root of the operator T and is denoted by T1/2. The existence of the square root allows one to determine the absolute value |T | = (T∗T)1/2 of the bounded operator T. For every bounded linear operator T: H → H there exists a unique partially isometric operator U: H → H such that T = U|T |, KerU = KerT. Such an equality is called a polar expansion of the operator T. The Schmidt operator is understood as the unitary multiplier of the polar expansion of a compact inverse operator, with the help of which E. Schmidt was the first to obtain the expansion of a compact and not-self-adjoint operator and introduced so-called s-numbers. This paper shows that the unitary symmetrizer of an operator differs only in sign from the adjoint Schmidt operator. The main result of the paper: if A is an invertible and compact operator, and S is a unitary operator such that the operator SA is self-adjoint, then the operator AS is also self-adjoint and the formula S = ±U∗ holds, where U is the Schmidt operator.
compact operator , normal operator , polar representation of operator , Schmidt expansion , Schmidt operator , square root of positive self-adjoint operator , symmetrizer , Unitary operator
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Akhmet Yassawi International Kazakh-Turkish University, Turkestan, Kazakhstan
Akhmet Yassawi International Kazakh-Turkish University
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