6,244 research outputs found
Explicit min–max polynomials on the disc
AbstractDenote by Πn+m−12≔{∑0≤i+j≤n+m−1ci,jxiyj:ci,j∈R} the space of polynomials of two variables with real coefficients of total degree less than or equal to n+m−1. Let b0,b1,…,bl∈R be given. For n,m∈N,n≥l+1 we look for the polynomial b0xnym+b1xn−1ym+1+⋯+blxn−lym+l+q(x,y),q(x,y)∈Πn+m−12, which has least maximum norm on the disc and call such a polynomial a min–max polynomial. First we introduce the polynomial 2Pn,m(x,y)=xGn−1,m(x,y)+yGn,m−1(x,y)=2xnym+q(x,y) and q(x,y)∈Πn+m−12, where Gn,m(x,y)≔1/2n+m(Un(x)Um(y)+Un−2(x)Um−2(y)), and show that it is a min–max polynomial on the disc. Then we give a sufficient condition on the coefficients bj,j=0,…,l,l fixed, such that for every n,m∈N,n≥l+1, the linear combination ∑ν=0lbνPn−ν,m+ν(x,y) is a min–max polynomial. In fact the more general case, when the coefficients bj and l are allowed to depend on n and m, is considered. So far, up to very special cases, min–max polynomials are known only for xnym,n,m∈N0
The Carath\'eodory-Fej\'er Interpolation Problems and the von-Neumann Inequality
The validity of the von-Neumann inequality for commuting - tuples of
matrices remains open for . We give a partial answer to
this question, which is used to obtain a necessary condition for the
Carath\'{e}odory-Fej\'{e}r interpolation problem on the polydisc
In the special case of (which follows from Ando's theorem as well), this
necessary condition is made explicit. An alternative approach to the
Carath\'{e}odory-Fej\'{e}r interpolation problem, in the special case of
adapting a theorem of Kor\'{a}nyi and Puk\'{a}nzsky is given. As a consequence,
a class of polynomials are isolated for which a complete solution to the
Carath\'{e}odory-Fej\'{e}r interpolation problem is easily obtained. A natural
generalization of the Hankel operators on the Hardy space of
then becomes apparent. Many of our results remain valid for any however, the computations are somewhat cumbersome for and are
omitted. The inequality , where
is the complex Grothendieck constant and
is due to Varopoulos. Here the
supremum is taken over all complex polynomials in variables of degree
at most and commuting - tuples of
contractions. We show that obtaining a slight improvement in the inequality of Varopoulos.
We show that the normed linear space has no isometric
embedding into complex matrices for any and
discuss several infinite dimensional operator space structures on it.Comment: This is my thesis submitted to Indian Institute of Science, Bangalore
on 20th July, 201
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