On some properties of the Gram operator
DOI:
https://doi.org/10.3842/umzh.v77i3.8685Keywords:
systems of subspaces, Gram operator, Hilbert space, orthogonal projectorAbstract
UDC 512.552.4
The $G$-construction allows us to obtain a system of subspaces using an abstract Gram operator. Any system of subspaces is unitarily equivalent to a system constructed by using this method. We propose another construction of a system of subspaces with the same property, provide criteria for the unitary equivalence of a pair of systems of subspaces in terms of the corresponding Gram operators, and study the problem of irreducibility of a system of subspaces in terms of the corresponding Gram operator. In addition, we provide an example illustrating how the obtained results apply to the study of a certain class of the systems of subspaces.
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