Questions tagged [operator-theory]

Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.

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Does this operator have a continuous, localized eigenfunction with negative eigenvalue?

I am looking at a class of operators $$ L[f](x)=af_{xxxx}-bf_{xx}+\frac{d}{dx}(\delta(x)f_x) $$ , a<0,b<0, on the real line, where $\delta$ is Dirac-delta. I am interested in ruling out the ...
mathamphetamine's user avatar
1 vote
1 answer
191 views

Dimension of commutant

Suppose that $A = M_n(\mathbb{C})$ be the algebra of $n*n$ matrices over $\mathbb{C}$. If com(A) = {$B \in M_n(\mathbb{C}); AB = BA$}, then what is the $dim(com(A))?$
Peg Leg Jonathan's user avatar
4 votes
0 answers
183 views

Double commutant of compact operators

So my question is straightforward. Let $\mathfrak{X}$ be a (complex, if necessary) Banach space and $K\colon\mathfrak{X}\to\mathfrak{X}$ a nonzero compact operator. Denote by $\mathcal{C}(K)$ the ...
Jack L.'s user avatar
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1 vote
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Reduced twisted $C^*$-algebra and twisted crossed product

Let $G$ be a discrete group. Is it possible to represent $C^*_r(G, \sigma)$, the reduced twisted group $C^*$-algebra as a twisted crossed product?
Peg Leg Jonathan's user avatar
3 votes
0 answers
632 views

The exponential derivative operator

Thank you very much for the interesting responses in my previous question The Quotient exponential operator. I have another complicated formula related to the previous one in the following $$ \exp\...
Adam Hammam's user avatar
0 votes
1 answer
191 views

The Quotient exponential operator

I have a question if you don't mind. I have the following quotient operator: $$\frac{1}{e^{d/dx}(f(x))}$$ Where $f$ is a smooth function on $R$. I would like to get rid of the denominator. IS there ...
Adam Hammam's user avatar
4 votes
1 answer
203 views

Does there exists a $C^*$-algebra corresponding to every Banach ternary algebra?

Let $V$ be a TRO (closed subspace of $B(H,K)$, closed under the product $xyz\to xy^*z$). Let $C(V)$, $D(V)$ denote the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^*V$ respectively. We define $A(...
Math Lover's user avatar
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8 votes
0 answers
307 views

Monotonicity of log determinant of Gaussian kernel matrix

Let \begin{equation} k({x},{y}) = \sigma \exp\left(-\frac{(x-y)^2}{2\theta^2}\right)\end{equation} be a squared-exponential (Gaussian) kernel, with $\sigma,\vartheta>0$. Consider, for a set of $N$ ...
Heinrich A's user avatar
4 votes
1 answer
266 views

Banach space with dual not a GT space

Let $X$ be a Banach space. A bounded linear map $u:X\to\ell_2$ is said to be $1$-summing if for all finite sequence $(x_i)\subseteq X$ there is a constant $C>0$ such that $\sum\|ux_i\|\leq C\sup\...
A beginner mathmatician's user avatar
4 votes
0 answers
71 views

Need reference of books/papers which deals with Ternary Banach Algebras

I'm interested in learning about ternary Banach Algebras ( mainly ideal theory and tensor product) Can someone please recommend me some papers/ books/ notes which deals with mentioned topics? Thank ...
Math Lover's user avatar
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2 votes
1 answer
104 views

Operators "building" linear independant sets

Let $E$ be a separable Banach space and let $T\in L(E,E)$. Is there a condition on $T$ ensuring that: $$ \mbox{$\{x_n\}_{n=1}^N\subseteq E$ is linearly independent} \Rightarrow \{T(x_n)\}_{n=1}^N\cup \...
TomCat's user avatar
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0 votes
1 answer
143 views

Inequality between matrix elements of positive self-adjoint operators

We have three positive semi-definite self-adjoint operators $\hat{A}_-$, $\hat{B}$, $\hat{A}_+$ on the Hilbert space $\mathcal{H}$. They are unbounded operators and satisfy the following inequality \...
Harry Song's user avatar
2 votes
0 answers
45 views

Additivity of squared Schatten $p$-norm with respect to spatial partition

Consider a Hilbert-Schmidt operator $A$ on $L^2(\mathbb R^d)$ with integral kernel $A(x,y)$. Let $\Omega\subset \mathbb R^d$ and $1_{\Omega}(x)$ denote its characteristic function as well as the ...
user271621's user avatar
1 vote
0 answers
47 views

Hypercylic operators have very typical cyclic vectors

Let $W$ be the Wiener measure on $C_0(\mathbb{R})$ and let $T\in L(C_0(\mathbb{R}),C_0(\mathbb{R}))$ be a hypercylic operator; i.e. there exists some $f \in C_0(\mathbb{R})$ such that $\{T^n(f)\}_{n=1}...
ABIM's user avatar
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11 votes
0 answers
334 views

Tauberian Theorem for 1-parameter groups of operators

The Wiener Tauberian Theorem gives condition on an $f\in L^1(\mathbb{R})$ such that the "induced 1-parameter family" $\{T_b(f)\}_{b\in \mathbb{R}}$ has a dense span in $L^1(\mathbb{R})$; ...
ABIM's user avatar
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4 votes
0 answers
145 views

Cyclic vectors for the translation operator

Let $b\in \mathbb{R}\neq 0$, and consider the translation operators: $$ \begin{align} T_b:C(\mathbb{R}) & \rightarrow C(\mathbb{R})\\ f &\mapsto f(\cdot + b). \end{align} $$ *Are there known ...
ABIM's user avatar
  • 5,019
0 votes
0 answers
72 views

Compact operators and projective tensor space

I know that the space of all the bounded linear maps between two Banach spaces, denoted by $L(X,Y)$, has a relationship with the projective tensor space of $X$ and $Y$, $$({X \widehat\otimes_{\pi} Y})...
math1298's user avatar
3 votes
0 answers
113 views

Schatten norm estimate of spatially truncated resolvent of Laplacian

Consider the operator $-\Delta$ on $L^2(\mathbb R^d)$. In my studies I stumbled upon operators of the form $$1_{\Gamma_n} (-\Delta -z)^{-1} 1_{\Gamma_m},$$ where $1_{\Gamma_m}$ denotes multiplication ...
user271621's user avatar
1 vote
0 answers
104 views

Self-adjointness of an operator

This is a problem from page 3 of Bourgain, Burq, and Zworski - Control for Schrödinger equations on 2-tori: rough potentials, the author claim that Hence $P = - \partial_x^2 + W$ defined on $C^\infty(...
Xin Fu's user avatar
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2 votes
0 answers
49 views

Determining a space of differentiability

I have a questions and maybe you are able to assist with this? Let us consider the space $X:=\mathrm{L}^2[0,\pi]$. On $X$ we consider the family of operators $(P(t,s))_{t\geq s}$ defined by $$ P(t,s)f:...
Daniella Dannell's user avatar
1 vote
1 answer
103 views

Relation between the solutions $v_t=Lv$ and $v_t=Av$ if $A$ is a relatively compact perturbation of the linear operator $L$

In a nutshell, here is my question. I read and know about the relation between the spectra of $L$ and $A$ if $A$ is a relatively compact perturbation of $L$. However, for my purpose, I am interested ...
Gateau au fromage's user avatar
2 votes
1 answer
211 views

Limit of $e^{-t(-\Delta)^\alpha}$ when $\alpha \to 1$

Let us consider the fractional heat semigroup $\left(e^{-t(-\Delta)^\alpha}\right)_{t\ge 0}$ for $\alpha\in (0,1)$ (the fractional power is taken in whole $\mathbb R^d$). Is there any result about the ...
Migalobe's user avatar
  • 395
3 votes
1 answer
201 views

Reference request: Baire's theorem for operator ranges

Let $F$ be a Banach space. A vector subspace $U \subseteq F$ is called an operator range if there exists a Banach space $E$ and a bounded linear mapping $T: E \to F$ such that $TE=U$. By a quotient ...
Jochen Glueck's user avatar
3 votes
1 answer
383 views

Inverse of block matrix

Let $V$ be a finite-dimensional vector space and consider the space $X=V\times V\times V\times V.$ Consider the block matrix $$A = \begin{pmatrix} A_1 & A_2 \\ A_2^* & -A_1\end{pmatrix}$$ ...
Kung Yao's user avatar
  • 192
3 votes
0 answers
123 views

Zygmund class, Schwartz class and Littlewood-Paley projection operators

I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions: Consider the Zygmund class of functions defined as ...
pureorapplied's user avatar
3 votes
0 answers
227 views

Hurwitz–Radon problem for $ \mathbb{Q} ^{n} $

What is the maximal number of orthogonal operators $ A _{1} , \dotsc, A _ {m} $ in $ \mathbb{Q}^{ n } $ satisfying the relations $ A_{i}^{2} = - I $ and $ A_{i}A_{j} + A_{j}A_{i} = 0 $ for $ i \neq ...
Sky's user avatar
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2 votes
0 answers
75 views

Convergence of random operators

I'm a statistician not versed in functional analysis and operator theory. I wish that I might not find a wrong place for my question. All my questions are trivial in the scalar time series case, but ...
metric's user avatar
  • 121
4 votes
1 answer
541 views

Derivative of trace

Consider two positive-semi definite matrices $T_1, T_2$ of unit trace. Let $T(\lambda):=T_1 + \lambda(T_2-T_1)$ be the convex combination of the two. We then study $f(\lambda) := \operatorname{tr}(T(\...
Sascha's user avatar
  • 506
0 votes
0 answers
109 views

Methods to find the spectrum of an operator

Suppose we have a bounded, self-adjoint operator $T$ on a set of functions $\mathcal{F}$. What kinds of methods are there to find the spectrum of $T$? Here is the setting I'm wondering about: consider ...
lady gaga's user avatar
  • 153
4 votes
1 answer
284 views

Uniform boundedness principle for almost surely converging sequence of operators

I'd like to do the following: I consider a separable Banach space $X$ with a probability measure $\mu$ on the Borel $\sigma$-algebra $\mathcal B(X)$. Additionally, I have a sequence of measurable, ...
Philipp Wacker's user avatar
10 votes
2 answers
1k views

Differentiation of functions over graphs

In short: There are various ways to define differentiation over a graph. I am trying to get the big picture, like a more complete and structured bestiary. Definitions. Let $G=(V,E)$ be a directed ...
Matthieu Latapy's user avatar
3 votes
0 answers
163 views

Eigenvalue estimates for kernel integral operator for Laplace kernel on unit-sphere in high-dimensions

Let $d$ be a large positive integer and let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $K_\gamma:S_{d-1} \times S_{d-1} \to \mathbb R$ be defined by $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
dohmatob's user avatar
  • 6,716
3 votes
1 answer
263 views

Opposite $C^*$ algebras

$\DeclareMathOperator\op{op}$Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $...
A beginner mathmatician's user avatar
2 votes
0 answers
475 views

Convergence of operator in norm resolvent sense and their eigenvectors

Let $\{T_n\}_{n=1}^\infty$ and $T$ be (unbounded) self-adjoint operators and $T_n\to T$ in norm resolvent sense, that is, for some $z\in \mathbb{C} \setminus \mathbb{R}$, $\|(zI- T_n)^{-1}- (zI- T)^{-...
user270619's user avatar
4 votes
0 answers
161 views

Boundary conditions for singular Sturm-Liouville problem (boundary behavior of eigenfunctions)

I am not at all an expert in Sturm-Liouville theory, but I ended up on the following Singular Sturm Liouville problem: \begin{equation}\label{1} (1) \ \ \ \ \ \ \ \ \ \ \ y''(t)+\frac{\theta'(t)}{\...
L. Proz's user avatar
  • 83
0 votes
0 answers
152 views

When is the heat semigroup Gibbs?

Defining the Laplacian on a region $Ω$ of $\mathbb{R}^d$ with Dirichlet boundary conditions, under what conditions on the region (or any other possible assumptions) is the semigroup it generates Gibbs,...
folouer of kaklas's user avatar
0 votes
0 answers
127 views

Certain decompositions of decomposable Banach spaces

Let $\mathcal{X}$ be a decomposable Banach space (i.e. a topological direct sum of infinite-dimensional subspaces, say $\mathcal{X}=\mathcal{A}\oplus\mathcal{B}$). Can one always obtain another ...
Jack L.'s user avatar
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0 votes
0 answers
142 views

Dual operator space

Suppose $E$ is an operator space and $E^*$ is the dual operator space. It is well known that the matricial norm structure on $E^*$ is given by the formula $\|[f_{ij}]_{i,j=1}^n\|_{M_n(E^*)}:=\sup\{\|...
A beginner mathmatician's user avatar
3 votes
1 answer
631 views

Operator norm of difference of matrix decompositions

This question is in part related to a question that I have already posed. Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}_A \...
Heinrich A's user avatar
0 votes
1 answer
60 views

Meaning of $K'[\cdot]$ when $K$ is an symmetric Onsager (matrix) operator

My question is from the section 5.2 of the monograph "Entropy Methods for Diffusive Partial Differential Equations" written by Ansgar J$\ddot{\text{u}}$ngel. To be specific, I do not know ...
Fei Cao's user avatar
  • 700
2 votes
1 answer
236 views

Trying to understand Haagerup tensor product $B(H)\otimes_{\rm h}B(K)$

I’m self reading Haagerup tensor product of operator spaces. Understanding it properly, I’m trying some examples. Let $H$ And $K$ be Hilbert space. Let $B(H)$ and $K(H)$ denotes the spaces of bounded ...
Math Lover's user avatar
  • 1,065
4 votes
1 answer
351 views

English translation of von Neumann's Algebra der Funktionaloperationen (1930)

Does anyone know if there exists an English translation of von Neumann's early work in operator theory, in particular the paper Zur Algebra der Funktionaloperationen und Theorie der normalen ...
lanf's user avatar
  • 295
0 votes
0 answers
67 views

Multiplication of a Riesz basis

Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1) \times L^2(0,1)$. My question is the following: If we multiply the basis by the matrix $e^{Mx}$, $x \in (0,1)$ where $M$ is a ...
Gustave's user avatar
  • 545
1 vote
0 answers
138 views

Haagerup tensor product

Let $H$ and $K$ be Hilbert spaces. Recall that a closed subspace $V\subset B(H,K)$ is called a ternary ring of operators(TRO) if $xy^*z \in V$ for all $x,y,z \in V$. Obviously a TRO is also a ...
Math Lover's user avatar
  • 1,065
3 votes
3 answers
574 views

Universal property of tensor products of bounded operators

Consider the tensor product of bounded operators. Does this tensor product satisfy the universal property of the tensor product, i.e., for any bilinear map $F: B(\mathcal H_1)\times B(\mathcal H_2)\to ...
Dominique Unruh's user avatar
2 votes
1 answer
131 views

Lifting theorem for n operators

I am aware of the following statement of the lifting theorem. For $i\in \{1,2\}$ let $B_i$ be a contraction on a Hilbert space $H_i$ and let $A_i$, acting on the Hilbert space $K_i$, be the minimal ...
Bhawna Bansal's user avatar
4 votes
0 answers
202 views

Spectral theorem for unbounded operators

Part of the Spectral theorem for unbounded operators states that if $A$ is a self adjoint unbounded operator and $B$ is a bounded operator such that $BA$ is contained in $AB$, then $B$ commutes with ...
Isaac's user avatar
  • 771
2 votes
0 answers
134 views

Extensions of symmetric unbounded operators

I saw it claimed that every symmetric operator on a Hilbert space $H$ can be extended to a self-adjoint operator on some potentially larger space K. But I seem to be able to prove from this that every ...
Isaac's user avatar
  • 771
1 vote
0 answers
127 views

A question on Villani's monograph "Hypocoercivity"

I can not figure out the appearance of the term $\int h_0\,d\mu$ in the statement of Theorem 35 above. Here are some background information: $L$ is an unbunded operator on a Hilbert space $\mathcal{H}^...
Fei Cao's user avatar
  • 700
3 votes
0 answers
318 views

Heat equation damps backward heat equation?

In a previous question on mathoverflow, I was wondering about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. ...
Sascha's user avatar
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