### Isospectral torus for banded Hessenberg matrices

A sequence of orthogonal polynomials (Pn(x))¥

n=0 on the real line gives rise to the well-known Jacobi matrix

J =

0

BBBB@

b0 a0 0

a

0 b1 a1

a

1

. . .

. . .

0 . . .

. . .

1

CCCCA

:

Let us call J periodic with period r if ak ak+r and bk bk+r for all k 0. In this case the zeros of Pn(x)

for n!¥ are asymptotically distributed on a set G R which is the disjoint union of at most r intervals

[?].

Conversely, for any such set G R, one can ask:

Is there a periodic Jacobi matrix associated to this set G?

If so, what is the structure of the 'isospectral manifold' formed by all such Jacobi matrices?

The answer to the rst question can be yes or no depending on a nice potential theoretic criterion. For the

second question, the structure of the isospectral manifold turns out to be a torus [1].

In this talk we study the generalization of these questions to the setting of banded Hessenberg matrices,

i.e., we replace the Jacobi matrix J by a matrix which has several non-vanishing diagonals in its lower

triangular part. We discuss the connection with Nikishin systems.

[1] B. Simon, Szego's Theorem and its Descendants, Princeton university press, 2010.

** This is a joint work with: A. Lopez-Garca.

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