Questions tagged [toeplitz-operators]
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33 questions
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Maximal eigenvalue of a real symmetric Toeplitz matrix
The $n×n$ matrix $A_n$ is defined by the elements $a_{ij}=n−|i−j|$.
\begin{bmatrix}
n & n-1 & n-2 & \cdots & 1\\
n-1 & n & n-1 & \cdots & 2\\
n-2 & n-1 & n &...
- 61
2
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1
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Show that $\|P(f\circ\varphi_{\lambda})-\widetilde{f}(\lambda)\|_p=\|P(f\circ\varphi_{\lambda}-\overline{P(\overline{f}\circ\varphi_{\lambda}}))\|_p.$
Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman ...
- 41
1
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0
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64
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Reference request for non-banded Toeplitz matrix
I want to know references that treat the property of eigenvalues and eigenstates of the non-banded Toeplitz matrix.
I mean for example, the Toeplitz matrix $A$ whose matrix element is given by $A_{ij}=...
- 11
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Reciprocal of a Fourier series (explicit formula for the zero mode?)
Consider a complex Fourier series
$$f(\varphi)=\sum_{m=-\infty}^\infty a_me^{im\varphi}$$
Its reciprocal (where it exists) also admits a Fourier series expansion
$$\frac{1}{f(\varphi)}=\sum_{m=-\infty}...
- 1,035
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92
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Proof of the isomorphism of the Toeplitz algebra and the algebra generated by the element and the relation
Please tell me where can I see the proof of this well-known fact?
enter image description here
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0
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What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^{\frac{n-1}{2}}$ when $n$ is odd?
Suppose that $n$ is odd. The eigen values/eigenvectors of the skew-circulant matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ are successfully computed in this post.
Q. What are ...
- 4,058
3
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1
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The eigenvalues of the matrix $\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$
What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ when $n$ is odd?
- 4,058
2
votes
1
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Monotonicity of the determinant of symmetric Toeplitz Matrices
For simplicity, i focus on a particular Toeplitz symmetric matrix, so let $A_{ij} = a^{|i-j|}$ for $i,j=1,\dots,n$ and $0<a<1$ be a Kac-Murdock-Szegő (KMS) matrix, e.g., for n=4
\begin{equation}
...
- 77
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214
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How to calculate Toeplitz-type determinant expansion?
We want to calculate next sum in different point in limit of large $N, N_f$.
Initial expression is expansion around $h=0$ by definition. And the answer is known . This is ($N_f/N= \kappa$)
$$
\lim_{N ...
1
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0
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67
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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 ...
- 243
1
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1
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265
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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^*...
- 9,853
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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 $|\...
- 121
7
votes
1
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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 ...
- 640
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2
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Eigenvalue density of a symmetric tridiagonal matrix
Let $A_n\in\mathbb{R}^{n\times n}$ be defined as
$$
A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...
- 2,712
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1
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C*-algebra of free monogenic inverse semigroup
Let us take the right shift operator $S$ acting on the Hilbert space $l^2(\mathbb{N})$. Consider the C*-algebra generated the operator
$
\begin{pmatrix}
S & 0 \\
0 & S^*
\end{pmatrix}
$ ...
- 493
4
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245
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On the sum of the first row of the inverse of a certain symmetric Toeplitz matrix
(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows:
$$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$
Let $...
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Reference for a text book on the Toeplitz operators $T:l^{\infty}(\mathbb Z, \mathbb R^2)→l^{\infty}(\mathbb Z, \mathbb R^2)$
We need basic reference on the Toeplitz operators $T:l^{\infty}(\mathbb Z, \mathbb R^2)→l^{\infty}(\mathbb Z, \mathbb R^2)$. Usually text books cover much more subtle case of $T:l^{\infty}(\mathbb Z_+,...
- 11
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2
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215
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$l^\infty$ spectrum of Toeplitz operator
We have the Toeplitz operator $T:l^{\infty}(Z, R^2) \to l^{\infty}(Z, R^2)$. We computed spectrum of $T$ on $l^2$ using its symbol (symbol is continuous function $\varphi(z)$ and eigenvalues of $\...
- 21
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2
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On the limit set of eigenvalues of banded Toeplitz Hessenberg matrices
Let $T_{n}(b)$ be the $n\times n$ Toeplitz matrix determined by the symbol
$$
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$. ...
- 2,188
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63
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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 ...
- 61
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0
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119
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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.
...
- 61
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0
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118
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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 ...
- 183
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0
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639
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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,...
- 3,873
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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\...
- 131
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5
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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 ...
- 1,418
1
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1
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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 = \{\...
- 228
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271
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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 ...
- 2,383
2
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1
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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 ...
- 291
1
vote
1
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312
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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
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1
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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. ...
- 20.2k
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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....
- 29.2k
4
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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 $...
- 2,044
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2
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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 ...
- 54.6k