## Toeplitz operators and related topics

#### Location

Cupples I Room 215

#### Start Date

7-18-2016 3:00 PM

#### End Date

18-7-2016 3:20 PM

#### Description

The set $AP$ of (Bohr) almost periodic functions is the closed subalgebra of $L_\infty(\mathbb R)$ generated by all the exponents $e_\lambda(x):=e^{i\lambda x}$, $\lambda\in\mathbb R$. An $AP$ factorization of an $n$-by-$n$ matrix function $G$ is its representation as a product $G=G_+\text{diag}[e_{\lambda_1},\ldots,e_{\lambda_n}]G_-,$ where $G_+^{\pm 1}$ and $G_-^{\pm 1}$ have all entries in $AP$ with non-negative (resp., non-positive) Bohr-Fourier coefficients. This is a natural generalization of the classical Wiener-Hopf factorization of continuous matrix-functions on the unit circle, arising in particular when considering convolution type equation on finite intervals. The state of $AP$ factorization theory as of the beginning of the century can be found in the monograph Convolution Operators and Factorization of Almost Periodic Matrix Functions'' by A. B\"ottcher, Yu. I. Karlovich and myself. In this talk I will describe, time permitting, further results obtained and problems still open.

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COinS

Jul 18th, 3:00 PM Jul 18th, 3:20 PM

Almost periodic factorization in the twenty first century

Cupples I Room 215

The set $AP$ of (Bohr) almost periodic functions is the closed subalgebra of $L_\infty(\mathbb R)$ generated by all the exponents $e_\lambda(x):=e^{i\lambda x}$, $\lambda\in\mathbb R$. An $AP$ factorization of an $n$-by-$n$ matrix function $G$ is its representation as a product $G=G_+\text{diag}[e_{\lambda_1},\ldots,e_{\lambda_n}]G_-,$ where $G_+^{\pm 1}$ and $G_-^{\pm 1}$ have all entries in $AP$ with non-negative (resp., non-positive) Bohr-Fourier coefficients. This is a natural generalization of the classical Wiener-Hopf factorization of continuous matrix-functions on the unit circle, arising in particular when considering convolution type equation on finite intervals. The state of $AP$ factorization theory as of the beginning of the century can be found in the monograph Convolution Operators and Factorization of Almost Periodic Matrix Functions'' by A. B\"ottcher, Yu. I. Karlovich and myself. In this talk I will describe, time permitting, further results obtained and problems still open.