Toral $(m,C)$-isometric multivariable operators

Abstract In recent years, various aspects of the problem related to the generalization of the class of m -isometries of commuting tuples of operators in Hilbert spaces have appeared in the literature. Let us mention, for example, $(n_{1},\ldots ,n_{p})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:math> -quasi- m -isometric tuples, $(m,C)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:math> -isometric tuples, toral m -isometries, and toral $(m, A)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:math> -isometries. In this paper, we discuss a problem closely related to this one. We introduce the class of toral- $(m,C)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:math> -isometries for a tuple of commuting operators related to a given conjugation operator C on a Hilbert space H . We present its basic properties and demonstrate a variety of results that extend some works related to multivariable operators.