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Barik, Sibaprasad; Bisai, Bappa

A generalization of Ando’s dilation, and isometric dilations for a class of tuples of \(q\)-commuting contractions. (English) Zbl 07885444

Complex Anal. Oper. Theory 18, No. 6, Paper No. 131, 29 p. (2024).
Summary: Given a bounded operator \(Q\) on a Hilbert space \(\mathcal{H}\), a pair of bounded operators \((T_1, T_2)\) on \(\mathcal{H}\) is said to be \(Q\)-commuting if one of the following holds: \[ T_1T_2 = QT_2T_1 \text{ or }T_1T_2 = T_2QT_1 \text{ or }T_1T_2 = T_2T_1Q. \] We give an explicit construction of isometric dilations for pairs of \(Q\)-commuting contractions for unitary \(Q\), which generalizes the isometric dilation of Ando (Acta Sci Math (Szeged) 24:88-90, 1963) for pairs of commuting contractions. In particular, for \(Q = qI_{\mathcal{H}}\), where \(q\) is a complex number of modulus 1, this gives, as a corollary, an explicit construction of isometric dilations for pairs of \(q\)-commuting contractions, which are well studied. There is an extended notion of \(q\)-commutativity for general tuples of operators and it is known that isometric dilation does not hold, in general, for an \(n\)-tuple of \(q\)-commuting contractions, where \(n \geq 3\). Generalizing the class of commuting contractions considered by Brehmer (Acta Sci Math (Szeged) 22:106-111, 1961), we construct a class of \(n\)-tuples of \(q\)-commuting contractions and find isometric dilations explicitly for the class.

MSC:

47A20 Dilations, extensions, compressions of linear operators
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47B38 Linear operators on function spaces (general)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
32A70 Functional analysis techniques applied to functions of several complex variables

Keywords:

\(Q\)-commuting contractions; \(q\)-commuting contractions; isometric dilation; Szegö positivity; Brehmer positivity; Hardy space
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References:

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