475 research outputs found
On the complete boundedness of the Schur block product
We give a Stinespring representation of the Schur block product, say (*), on
pairs of square matrices with entries in a C*-algebra as a completely bounded
bilinear operator of the form: A:=(a_{ij}), B:= (b_{ij}): A (*) B :=
(a_{ij}b_{ij}) = V* pi(A) F pi(B) V, such that V is an isometry, pi is a
*-representation and F is a self-adjoint unitary. This implies an inequality
due to Livshits and two apparently new ones on diagonals of matrices. ||A (*)
B|| \leq ||A||_r ||B||_c operator, row and column norm; - diag(A*A) \leq A* (*)
A \leq diag(A*A), and for all vectors f, g: | |^2 \leq < diag(AA*)
g, g> .Comment: 10 p, revised, expanded and to appear in Proc. AM
Resolution of Peller's problem concerning Koplienko-Neidhardt trace formulae
A formula for the norm of a bilinear Schur multiplier acting from the
Cartesian product of two copies of the
Hilbert-Schmidt classes into the trace class is established in
terms of linear Schur multipliers acting on the space of
all compact operators. Using this formula, we resolve Peller's problem on
Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice
continuously differentiable function with a bounded second derivative, a
self-adjoint (unbounded) operator and a self-adjoint operator such that
f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $
Quasi-nilpotency of generalized Volterra operators on sequence spaces
We study the quasi-nilpotency of generalized Volterra operators on spaces of
power series with Taylor coefficients in weighted spaces
. Our main result is that when an analytic symbol is a multiplier for a
weighted space, then the corresponding generalized Volterra operator
is bounded on the same space and quasi-nilpotent, i.e. its spectrum is
This improves a previous result of A. Limani and B. Malman in the case
of sequence spaces. Also combined with known results about multipliers of
spaces we give non trivial examples of bounded quasi-nilpotent
generalized Volterra operators on .
We approach the problem by introducing what we call Schur multipliers for
lower triangular matrices and we construct a family of Schur multipliers for
lower triangular matrices on related to summability
kernels. To demonstrate the power of our results we also find a new class of
Schur multipliers for Hankel operators on , extending a result of E.
Ricard.Comment: 14 pages; The main theorems are the same as in v1, the presentation
of the material though, has changed drasticall
Complete boundedness of the Heat Semigroups on the von Neumann Algebra of hyperbolic groups
We prove that defines a
completely bounded semigroup of multipliers on the von Neuman algebra of
hyperbolic groups for all real number . One ingredient in the proof is the
observation that a construction of Ozawa allows to characterize the radial
multipliers that are bounded on every hyperbolic graph, partially generalizing
results of Haagerup--Steenstrup--Szwarc and Wysocza\'nski. Another ingredient
is an upper estimate of trace class norms for Hankel matrices, which is based
on Peller's characterization of such norms.Comment: v2: 28 pages, with new examples, new results, motivations and
hopefully a better presentatio
Operator Lipschitz functions on Banach spaces
Let , be Banach spaces and let be the space of
bounded linear operators from to . We develop the theory of double
operator integrals on and apply this theory to obtain
commutator estimates of the form for a large class of functions
, where , are scalar type
operators and . In particular, we establish this
estimate for and for diagonalizable operators on and
, for and , and for . We also
obtain results for . We also study the estimate above in the setting
of Banach ideals in . The commutator estimates we derive hold
for diagonalizable matrices with a constant independent of the size of the
matrix.Comment: Final version published in Studia Mathematica, with some minor
change
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