# Questions tagged [toeplitz-operators]

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26
questions

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### Monotonicity of the determinant of symmetric Toeplitz Matrices

For simplicity, i focus on a particular Toeplitz symmetric matrix, so let $A_{ij} = a^{|i-j|}$ for $i,j=1,\dots,n$ and $0<a<1$ be a Kac-Murdock-Szegő (KMS) matrix, e.g., for n=4
\begin{equation}
...

0
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0
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### How to calculate Toeplitz-type determinant expansion?

We want to calculate next sum in different point in limit of large $N, N_f$.
Initial expression is expansion around $h=0$ by definition. And the answer is known . This is ($N_f/N= \kappa$)
$$
\lim_{N ...

1
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0
answers

62
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### Toeplitz operators for other measures then Lebegue

In the standard setting there is a lot known about Toeplitz operators i.e that the compression of a multiplication operator restricted to the Hardy space.
Are there any results when one has a ...

1
vote

1
answer

101
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### Proof of universality of Toeplitz algebra

It is well-known that the Toeplitz algebra $\mathcal{T}$ (that I view as concrete subalgebra of $\mathbb{B}(\ell^2(\mathbb{N})$) is the universal algebra generated by an isometry, that is, for any $C^*...

2
votes

0
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### Largest eigenvalue scaling in a certain Kac-Murdoch-Szegö matrix

I'm looking at $N\times N$ matrices $M_N$ with elements
$$M_N=\left( \rho^{|i-j|} \right)_{i,j=1}^N,$$
where $\rho$ is a complex number of unit modulus.
These matrices with $\rho\in\mathbb R$ and $|\...

7
votes

1
answer

1k
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### Determinant and Inverse of a Toeplitz matrix

Let $T(n,k)$ be a $n \times n$ symmetric Toeplitz matrix, where all the entries of first $k$ super-diagonal (and sub-diagonal), last $k-1$ super-diagonal (and sub-diagonal) are ones, and rest of the ...

4
votes

2
answers

179
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### Eigenvalue density of a symmetric tridiagonal matrix

Let $A_n\in\mathbb{R}^{n\times n}$ be defined as
$$
A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...

0
votes

1
answer

143
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### C*-algebra of free monogenic inverse semigroup

Let us take the right shift operator $S$ acting on the Hilbert space $l^2(\mathbb{N})$. Consider the C*-algebra generated the operator
$
\begin{pmatrix}
S & 0 \\
0 & S^*
\end{pmatrix}
$ ...

4
votes

0
answers

213
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### On the sum of the first row of the inverse of a certain symmetric Toeplitz matrix

(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows:
$$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$
Let $...

1
vote

0
answers

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### Reference for a text book on the Toeplitz operators $T:l^{\infty}(\mathbb Z, \mathbb R^2)→l^{\infty}(\mathbb Z, \mathbb R^2)$

We need basic reference on the Toeplitz operators $T:l^{\infty}(\mathbb Z, \mathbb R^2)→l^{\infty}(\mathbb Z, \mathbb R^2)$. Usually text books cover much more subtle case of $T:l^{\infty}(\mathbb Z_+,...

2
votes

2
answers

188
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### $l^\infty$ spectrum of Toeplitz operator

We have the Toeplitz operator $T:l^{\infty}(Z, R^2) \to l^{\infty}(Z, R^2)$. We computed spectrum of $T$ on $l^2$ using its symbol (symbol is continuous function $\varphi(z)$ and eigenvalues of $\...

1
vote

2
answers

102
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### On the limit set of eigenvalues of banded Toeplitz Hessenberg matrices

Let $T_{n}(b)$ be the $n\times n$ Toeplitz matrix determined by the symbol
$$
b(z)=\frac{1}{z}+\sum_{j=0}^{k}a_{j}z^{j}
$$
where $k\in\mathbb{N}$ and $a_{0},\dots,a_{k}\in\mathbb{R}$, $a_{k}\neq0$. ...

2
votes

0
answers

60
views

### Riemann theta function with asymptotically large Toeplitz Matrix

As a follow up to
How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently
Suppose that $M$ is a large Toeplitz matrix. With a suitable scaling $K^{-n}$ for some $K$, what will the Riemann ...

1
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0
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102
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### inverse of asymptotic Toeplitz matrix with band limited associated function

I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation.
...

1
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0
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113
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### Asymptotic determinant of $2\times 2$ Toeplitz matrix

The problem that I am dealing with is to compute the determinant of a $2\times 2$ Toeplitz matrix[1] (in general I would like to generalize to a more general case, but let's consider the easiest case ...

1
vote

0
answers

578
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### On triangular Toeplitz matrices

Let $R(x)$ be the upper triangular Toeplitz matrix with first row $x$, so that $R_{ik}=x_{k-i+1}$ if $i\le k$ and $R_{ik}=0$ otherwise. Let $N(n)$ be the smallest number $N$ such that there exist $u_j,...

3
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0
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### Is there a possibility to find the toeplitz matrix that has the nearest column space to a given matrix?

As is said, if we have a known matrix $D \in \mathbb{R}^{N \times k}$ with $k<N$, is there a way that we can find a toeplitz matrix $D_T \in \mathbb{R}^{N \times k}$, which satisfies
$$ D_T=\arg\...

9
votes

5
answers

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### The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$.
Is there any ...

1
vote

1
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162
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### Infinite Real Symmetric Toeplitz Matrix Reference

I am looking for a good starting point (book or articles) for studying Toeplitz matrices. Specifically as mentioned in the title, I am most interested in the case where they are of the form
$$A = \{\...

0
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0
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258
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### L1-regularized Least Squares on a matrix with Toeplitz Blocks (not block-Toeplitz)

I am trying to speed up a sparse signal recovery algorithms.
My sensing matrix is a set of Toeplitz Blocks, M = [T1,T2,T3,...,Tk]
The objective is min ||Mx - b||_2^2 + ||x||1
What I'm actually ...

2
votes

1
answer

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### Inverse of an AR(1) or Laplacian (?) or Kac-Murdock-Szegö matrix

My current problem involves having an exact (symbolic) inverse of a scaled AR(1) matrix for $n$-dimension. (I don't know what this matrix would be called in general; I'm sure it is used often.) This ...

1
vote

1
answer

293
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### Invertibility of frame/sampling operator on Bargmann-Fock spaces

Let $F_\alpha ^p (\mathbb{C}^n)$ for $1 < p < \infty$ and $\alpha > 0$ be the Bargmann-Fock space defined as the Banach space of entire functions $f$ such that $f(\cdot) e^{- \frac{\alpha}{2} ...

5
votes

1
answer

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### annihilator/common left multiple of matrix polynomials

Let $A_{n,d}$ be the space of polynomials of degree $d$ or less whose coefficients are real $n\times n$ matrices --- or, if you prefer, the space of matrices whose entries are degree-$d$ polynomials. ...

4
votes

0
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257
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### Can one pose a Toeplitz index problem associated to a discrete group?

Before posing my question, let me provide a little background since the Wikipedia page on this stuff is sorely lacking.
Let's start with the classical case of the Toeplitz index problem on the circle....

4
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0
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### Convolutions and Toeplitz Operators

Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$.
Let be $...

8
votes

2
answers

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### Where can I learn about (the asymptotics of) Toeplitz operators?

Toeplitz operators provide a natural language with which to do geometric quantization. I don't want to really understand them, and I don't need them in full generality. I'm looking for some ...