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Smooth operators in the commutant of a contraction

Pascale Vitse (2003)

Studia Mathematica

For a completely non-unitary contraction T, some necessary (and, in certain cases, sufficient) conditions are found for the range of the H calculus, H ( T ) , and the commutant, T’, to contain non-zero compact operators, and for the finite rank operators of T’ to be dense in the set of compact operators of T’. A sufficient condition is given for T’ to contain non-zero operators from the Schatten-von Neumann classes S p .

Sobczyk's theorem and the Bounded Approximation Property

Jesús M. F. Castillo, Yolanda Moreno (2010)

Studia Mathematica

Sobczyk's theorem asserts that every c₀-valued operator defined on a separable Banach space can be extended to every separable superspace. This paper is devoted to obtaining the most general vector valued version of the theorem, extending and completing previous results of Rosenthal, Johnson-Oikhberg and Cabello. Our approach is homological and nonlinear, transforming the problem of extension of operators into the problem of approximating z-linear maps by linear maps.

Standard commuting dilations and liftings

Santanu Dey (2012)

Colloquium Mathematicae

We identify how the standard commuting dilation of the maximal commuting piece of any row contraction, especially on a finite-dimensional Hilbert space, is associated to the minimal isometric dilation of the row contraction. Using the concept of standard commuting dilation it is also shown that if liftings of row contractions are on finite-dimensional Hilbert spaces, then there are strong restrictions on properties of the liftings.

Standard dilations of q-commuting tuples

Santanu Dey (2007)

Colloquium Mathematicae

We study dilations of q-commuting tuples. Bhat, Bhattacharyya and Dey gave the correspondence between the two standard dilations of commuting tuples and here these results are extended to q-commuting tuples. We are able to do this when the q-coefficients q i j are of modulus one. We introduce a “maximal q-commuting subspace” of an n-tuple of operators and a “standard q-commuting dilation”. Our main result is that the maximal q-commuting subspace of the standard noncommuting dilation of a q-commuting...

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