# 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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### Inversion of integral transforms

We have the following theorem statement of this Paper "An inversion formula for a Class of integral transforms, 1975, C.Nasim" page 8. Let $\phi(x)= \frac{1}{x} \int_{0}^{x} \psi(t) dt$, ...
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### 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 ...
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### 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 ...
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### 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, ...
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### Determine the norm of a continuous linear operator $T:L^1[a,b]\to L^1[a,b]$ [closed]

I have encountered the following exercise: Let $K(x,y)$ be a measurable function on $[a,b]\times [a,b]$. The function $I:y\in [a,b] \mapsto \int_a^b |K(x,y)|\, \text{d}x \in [0,+\infty]$ belongs ...
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### 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 ...
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### 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 $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
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### 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)}{\...
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### 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,...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### Mapping properties of backward and forward heat equation

In a previous question on mathoverflow, I asked about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. The ...
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### Showing the existence of a right-inverse in a von Neumann probability space

Disclaimer: This is my first post here on Overflow as opposed to the "normal" forum, so if this question is too elementary for this forum, I'd appreciate y'all letting me know. I posted it ...
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