# 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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62 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 ...

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147 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 ...

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52 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 ...

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232 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(\...

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32 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 ...

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84 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, ...

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98 views

### 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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224 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 ...

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41 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 $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...

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126 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 $...

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55 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)^{-...

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96 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)}{\...

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123 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,...

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111 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 ...

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125 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\{\|...

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76 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 \...

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17 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 ...

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120 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 ...

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162 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 ...

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52 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 ...

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106 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 ...

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155 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 ...

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86 views

### lifting theorem for n case

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 ...

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128 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 ...

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74 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 ...

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108 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}^...

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284 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. ...

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122 views

### 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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20 views

### A problem related to Hardy space

I am having problems while proving this lemma.
Lemma: For $0<p\leq \infty$, we have $\overline{C_c(T)}=\Bbb{T}_{q,p,0}$.
Where, ${C_c(T)}=$ the set of all functions $f:T\to \Bbb{C}$ such that $M_p(...

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256 views

### Backward heat equation and forward perturbed heat equation well posed?

I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...

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154 views

### Reference for “Every compact quasinilpotent operator is the limit of nilpotent ones”

It was mentioned on Page 916 Problem 7 of Halmos's "Ten Problems in Hilbert space" that there is a proof for "Every compact quasinilpotent operator is the limit of nilpotent ones" ...

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88 views

### Strongly continuous semigroups on weighted $\ell^1$ space

Let $x=(x_i)$ be a sequence in $\ell^1$ such that all $x_i>0.$
Let $T(t):\ell^1 \rightarrow \ell^1$ be a strongly continuous semigroup of, i.e. $t \mapsto T(t)y$ is continuous for every $y \in \ell^...

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63 views

### 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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91 views

### Example of a ‘weak’ ideal which is not an ideal?

For Hilbert spaces $H$ and $K$, let $V=B(H,K)(H \neq K)$. A sub space $I$ of $V$ is called an ideal of $V$ if $$IV^*V+VV^*I \subset I$$ and $I$ is called weak ideal of $V$ If $$\text{span}\{IV^*V-VV^*...

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242 views

### Convergence of $\exp(tQ)$ in operator norm as $t\rightarrow\infty$

This is not a homework problem, so I am not sure whether this has a "good" answer or not. I came up with this question when I am now learning functional analysis and wonder whether my "...

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154 views

### How can I prove that $(I+\lambda G)$ is invertible, where $G$ is the Green function of an elliptic operator?

How can I prove that the operator $$(I+\lambda G)$$ is invertible, where $\lambda >0$ and $G$ is the Green function of an elliptic operator $A$ in a bounded domain $\Omega$?
$\Omega$ can be very ...

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57 views

### Show that the Laplacian on these domains is isospectral

Let $\Omega_i\subseteq\mathbb R^d$ be bounded and open, $A_i$ denote the weak Laplacian with domain $\mathcal D(A):=\{u\in H_0^1(\Omega_i):\Delta u\in L^2(\Omega_i)\}$ on $L^2(\Omega_i)$ and $$T_i(t)f:...

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52 views

### Proximity operator of lower semi-continuous and convex functions pre-composed with norm

Let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a some-where finite, lower semi-continuous, increasing, and convex function. It is easy to verify that the function $\Phi:\mathbb{R}^n\...

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88 views

### Series and solution of $-\Delta u + \lambda u = f(x)$

Consider a bounded smooth set $\Omega \subset \mathbb R^n$ (for example, we can take a ball). Can we write down the solution of
\begin{align*}
-\Delta u(x) + \lambda u(x) &= f(x), & x \in \...

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199 views

### When do positive operators have eigenvalues?

Let $B$ be a complex Banach lattice and let $T : B \to B$ be a positive operator. Are there any conditions that ensure that $T$ has an eigenvalue? I am interested in particular in non-compact ...

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### Sum of positive self-adjoint operator and an imaginary “potential”: literature request

To keep things simple, let us consider the following: $L$ is a positive, unbounded S.A. operator on $L_2(\mathbb{R},f(x))$, where $f(x)$ is a Gaussian. Assume that we know the spectrum and ...

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102 views

### Is there a discrete Schrödinger operator with empty spectrum?

A relatively well-known example of (continuous) Schrödinger operator with empty spectrum is the complex Airy operator on the line, i.e., the operator acting on $L^{2}(\mathbb{R})$ given by the ...

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70 views

### Special case of Elliott's Theorem

Let $A$ and $B$ be unital $AF$-algebra. By Elliott's theorem we know that if there an order isomorphism $\psi: K_0(A) \rightarrow K_0(B)$ with $\psi([1_{A}]) = [1_{B}]$, then there exists an ...

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131 views

### $\tau (p) = \tau (q)$ for all normalized traces does not imply $p \sim q$

Could you give an example of a unital simple $C^*$-algebra that $\tau (p) = \tau (q)$ for all normalized traces does not imply $p \sim q$?

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67 views

### On strong convergence versus weak in operator topology and semi-continuity of the spectrum

Let $\mathbb H$ be a Hilbert space and let $\mathcal B(\mathbb H)$ be the Banach algebra of bounded operators on $\mathbb H$. Let $(A_k)_{k\ge 1}$ be a sequence in $\mathcal B(\mathbb H)$.
$\bullet$ ...

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74 views

### Functional calculus for “pre-linear” regular operators on a Hilbert module

Let $E$ be a Hilbert module over a $C^*$-algebra $A$. Let $T\colon E\to E$ be a densely defined, unbounded $A$-linear operator. (In particular, the initial domain of $T$ is an $A$-submodule of $E$.) ...

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634 views

### What's the matrix of logarithm of derivative operator ($\ln D$)? What is the role of this operator in various math fields?

Babusci and Dattoli, On the logarithm of the derivative operator, arXiv:1105.5978, gives some great results:
\begin{align*}
(\ln D) 1 & {}= -\ln x -\gamma \\
(\ln D) x^n & {}= x^n (\psi (n+1)-\...

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50 views

### Stability of densly defined $C_{0}$-semigroup

Let $(S(t))_{t \geq 0}$ be a $C_{0}$-semigroup on $H$ where $H$ is a Hilbert space. Suppose that $(S(t))_{t \geq 0}$ satisfies the following estimate on a dense subspace on $H$
$$||S(t)x||_H \leq e^{-...

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79 views

### what is the geometry interpretation of fractional differintegral?

By some analogy, the integral and differential can be extend to factorial differintegral, see https://en.wikipedia.org/wiki/Differintegral
My question is, what is the he geometry interpretation of ...