36 research outputs found
Extensions of positive definite functions on amenable groups
Let be a subset of a amenable group such that and
. The main result of the paper states that if the Cayley graph of
with respect to has a certain combinatorial property, then every positive
definite operator-valued function on can be extended to a positive definite
function on . Several known extension results are obtained as a corollary.
New applications are also presented
Redheffer Products and Characteristic Functions
AbstractFuhrmann [Israel J. Math.16 (1973), 162-176], and subsequently Ball and Lubin [Pacific J. Math.63 (1976), 309-324] have studied a class of perturbations of completely nonunitary contractions. We extend their results concerning the computation of the characteristic function by using the "Redheffer product" machinery [J. Math. Phys.39 (1960), 269-286]. This has been familiar to system theory experts for many years and has been recently revived by Foias and Frazho ["The Commutant Lifting Approach to Interpolation Problems," Birkhäuser, Basel, 1990] to obtain alternate proofs in the theory of intertwining dilations of contractions on a Hilbert space. The proof obtained is conceptually surprisingly simple. An application is the recapture, from a point of view different from the original one, of a result concerning de BrangesⲠspaces, which have received renewed attention in recent years
Two remarks about nilpotent operators of order two
We present two novel results about Hilbert space operators which are
nilpotent of order two. First, we prove that such operators are indestructible
complex symmetric operators, in the sense that tensoring them with any operator
yields a complex symmetric operator. In fact, we prove that this property
characterizes nilpotents of order two among all nonzero bounded operators.
Second, we establish that every nilpotent of order two is unitarily equivalent
to a truncated Toeplitz operator.Comment: 7 pages. To appear in Proceedings of the AM
Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor
It is known that the eigenvalues of selfadjoint elements a,b,c with a+b+c=0
in the factor R^omega (ultrapower of the hyperfinite II1 factor) are
characterized by a system of inequalities analogous to the classical Horn
inequalities of linear algebra. We prove that these inequalities are in fact
true for elements of an arbitrary finite factor. A matricial (`complete') form
of this result is equivalent to an embedding question formulated by Connes.Comment: 41 pages, many figure