3,983 research outputs found

### Minimal representations of unitary operators and orthogonal polynomials on the unit circle

In this paper we prove that the simplest band representations of unitary
operators on a Hilbert space are five-diagonal. Orthogonal polynomials on the
unit circle play an essential role in the development of this result, and also
provide a parametrization of such five-diagonal representations which shows
specially simple and interesting decomposition and factorization properties. As
an application we get the reduction of the spectral problem of any unitary
Hessenberg matrix to the spectral problem of a five-diagonal one. Two
applications of these results to the study of orthogonal polynomials on the
unit circle are presented: the first one concerns Krein's Theorem; the second
one deals with the movement of mass points of the orthogonality measure under
monoparametric perturbations of the Schur parameters.Comment: 31 page

### Matrix orthogonal polynomials whose derivatives are also orthogonal

In this paper we prove some characterizations of the matrix orthogonal
polynomials whose derivatives are also orthogonal, which generalize other known
ones in the scalar case. In particular, we prove that the corresponding
orthogonality matrix functional is characterized by a Pearson-type equation
with two matrix polynomials of degree not greater than 2 and 1. The proofs are
given for a general sequence of matrix orthogonal polynomials, not necessarily
associated with an hermitian functional. However, we give several examples of
non-diagonalizable positive definite weight matrices satisfying a Pearson-type
equation, which show that the previous results are non-trivial even in the
positive definite case.
A detailed analysis is made for the class of matrix functionals which satisfy
a Pearson-type equation whose polynomial of degree not greater than 2 is
scalar. We characterize the Pearson-type equations of this kind that yield a
sequence of matrix orthogonal polynomials, and we prove that these matrix
orthogonal polynomials satisfy a second order differential equation even in the
non-hermitian case. Finally, we prove and improve a conjecture of Duran and
Grunbaum concerning the triviality of this class in the positive definite case,
while some examples show the non-triviality for hermitian functionals which are
not positive definite.Comment: 49 page

### An extension of the associated rational functions on the unit circle

A special class of orthogonal rational functions (ORFs) is presented in this
paper. Starting with a sequence of ORFs and the corresponding rational
functions of the second kind, we define a new sequence as a linear combination
of the previous ones, the coefficients of this linear combination being
self-reciprocal rational functions. We show that, under very general conditions
on the self-reciprocal coefficients, this new sequence satisfies orthogonality
conditions as well as a recurrence relation. Further, we identify the
Caratheodory function of the corresponding orthogonality measure in terms of
such self-reciprocal coefficients.
The new class under study includes the associated rational functions as a
particular case. As a consequence of the previous general analysis, we obtain
explicit representations for the associated rational functions of arbitrary
order, as well as for the related Caratheodory function. Such representations
are used to find new properties of the associated rational functions.Comment: 27 page

### One-dimensional quantum walks with one defect

The CGMV method allows for the general discussion of localization properties
for the states of a one-dimensional quantum walk, both in the case of the
integers and in the case of the non negative integers. Using this method we
classify, according to such localization properties, all the quantum walks with
one defect at the origin, providing explicit expressions for the asymptotic
return probabilities at the origin

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