# 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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### Diagonalizing selfadjoint operator on core domain

Let $A$ be a densely-defined, positive, self-adjoint operator with compact resolvent on a Hilbert space $H$. Then, $\text{Range}(1+A)=H$ and there is a basis for $H$ consisting of eigenvectors of $1+A$...

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### Operator norms or seminorms on Krein spaces

According to this this reference, we have the definition of a Krein space
I'm currently studying this kind of spaces and came across the concept of the $J$-norm or $J$-seminorm of an operator. I ...

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### Weakly compact operators into $c_0$ and other separable spaces

A Banach space $E$ is called Grothendieck if every weak* convergent sequence in the dual space $E^*$ is weakly convergent. A typical example of a Grothendieck space is $\ell_\infty$. Diestel proved ...

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### On the spectrum of Fokker–Planck with linear drift

The paper by Liberzon and Brockett, "Spectral analysis of Fokker–Planck and related operators arising from linear stochastic differential equations." SIAM Journal on Control and ...

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### Operator norm of linear functional $\varphi \mapsto \int_\Omega f\varphi$ with respect to different norms

Let $\Omega \subseteq \mathbb{R}^n$ be open. For some $f \in L^2(\Omega)$ consider the continuous linear functional $$T \colon C^\infty_c(\Omega) \to \mathbb{R}, \qquad T(\varphi) := \int_\Omega f \...

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### Infinite dimensional matrix solvability

In order to solve a boundary problem for a conducting hemisphere, the following matrix equation arises (derived from the boundary condition on the curved part of the surface) where we must solve for ...

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### When integrating by part produces a singularity

I'm currently interesting in the following operator:
$$O[f](x):=f(x)-2xe^{x^2}\int_x^{+\infty}dt \, e^{-t^2} f(t)$$ for all $x\in\mathbb{R}$ and $f$ smooth and decaying at infinity fast enough with ...

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### The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras

I'm currently reading the paper The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras and having difficulty in understanding the proof of Proposition $4.5$ from the paper.
Let $A$ ...

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### Is $I-S$ in my attempt of Fredholm alternative injective?

Let $E$ be a Banach space. Let $\mathcal K(E)$ be the space of all compact (bounded linear) operators from $E$ to $E$. For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. Let ...

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### Is $\Lambda:= \pi_2 \circ \pi_1:E \to L$ surjective?

Let $E$ be a Banach space. For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. Let $I:E \to E$ be the identity map. Let $T:E \to E$ be a compact (bounded linear) operator. ...

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### What is the status of The Halmos Similarity Problem?

What is the general status of "The Halmos Similarity Problem"(HSP) in Operator theory?For What conditions ,HSP has been solved?

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### Given $\sigma(AB-BA) = \{0\}$, what can be said about $\sigma(A)$ and $\sigma(B)$?

Let $\mathcal H$ be a separable Hilbert space, and $\mathfrak B(\mathcal H)$ denote the algebra of bounded linear operators on $\mathcal H$. Furthermore, let $A,B \in \mathfrak B(\mathcal H)$ be two ...

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### A subalgebra of $B(H)$ which does not contain a commutator element

Is there a commutative subalgebra $A\subset B(H)$ containing the 1 dimensional scalars with the following property:
The algebra $A$ has trivial intersection with the set of commutator ...

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### Is the weighted shift strong frequently hypercyclic?

One sided Shift
Let be $M$ separable metric space. Consider $X=M^{\mathbb{N}}$ the sequence space equipped with the product metric $d(x,y)=\sum_{i=1}^\infty |x_i-y_i|/2^i$ . Define the shift map $\...

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### On a compact operator in the plane

Let $\Omega \subset \mathbb R^2$ be a bounded domain with a smooth boundary. Let $$\bar{\partial}= \frac{1}{2} ( \partial_{x^1} + i \,\partial_{x^2}),$$
and let $G: L^2(\Omega)\to H^2(\Omega)$ be the ...

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### Compactly supported continuous functions as a Tomita algebra

Let $G$ be a locally compact group with modular function $\delta_G$ and consider $\mathcal{K}(G)$, the set of compactly supported continuous functions $G\to \mathbb{C}$, with the $*$-algebra structure ...

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### Does ${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\alpha_2 b ))$ imply that $l(a)l(b)=r(a)r(b)=0$?

Let $H$ be a Hilbert space and let $a,b\in B(H)$ be such that
$${\rm Tr}(l(a))={\rm Tr}(r(a))<\infty , {\rm Tr}(l(b))={\rm Tr}(r(b))<\infty,$$
$${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\...

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### Identities for operators in group algebras

Let C[G] be a group algebra for (typically) infinite noncommutative group G.
fix f,g -- functions $f,g : C[G]\times C[G] \to C[G]$. Let us consider the family of operators on $C[G]$ such that for the ...

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### approximation of a Feller semi-group with the infinitesimal generator

Let $T_t$ a Feller semigroup (see this) and let $(A,D(A))$ its infinitesimal generator.
If A is a bounded operator it is easy to show that the Feller semi-group is $e^{tA}$.
Is this formula always ...

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### A possible generalization of Pitt's theorem

Inspired by Pitt's theorem and this post we ask the following question:
First we remind the Pitt's theorem and also introduce a particular Banach space as averaging of $\ell^p$ spaces for $1\leq p ...

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### Property of pushouts in the category of unital $C^{\ast}$-algebras

Let $A$ be a unital $C^{\ast}$-algebra and $\{ f_i: A \rightarrow A_i \}_i$ a finite collection of morphisms of unital $C^{\ast}$-algebras, such that the associated map $A \rightarrow \prod_i A_i$ is ...

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### Higher theory o $C^{\ast}$-algebras and the $C^{\ast}$-algebra of a $\infty$-groupoid

Has someone already worked out what would be the infinity categorical analogue of the category of $C^{\ast}$-algebras? Given a groupoid $G$ we can associate a $C^{\ast}$-algebra $C^* (G)$, can we do ...

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### Convergence of inverse operator with projections

Let $X$ be a separable Hilbert space, and let $(e_i)_{i=1}^\infty$ be an orthonormal basis of $X$. For each $n\in \mathbb{N}$, let $X_n$ be the subspace spanned by $(e_i)_{i=1}^n$, and consider the ...

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### In the proof of Neural Tangent Kernel stays constant in infinite width limit, why the norm of the dual mapping operator equals operator norm of kernel

For a fixed distribution $p^{in}$ on the input space $ \mathbb{R}^{n_0}$,
consider a function space $\mathcal{F}$ defined as $\{{f: \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_L}}\}$.
On this space, ...

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### Duality of projective and injective tensor product

I want a reference of the following statement which I think is true. Let $X$ and $Y$ be Banach spaces with $X$ finite dimensional. Then $(X\otimes_\epsilon Y)^*$ is isometrically isomorphic to $(X^*\...

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### Elements that commute with $1$ in the pushout of a $C^{\ast}$-algebra

Suppose $B$ and $C$ are commutative unital $C^{\ast}$-algebras with $B \subseteq C$ (unital). Let $c$ be an element of $C$ such that $c \ast 1 = 1 \ast c$ in the pushout (in the category of ...

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### Takesaki II "Connes cocycle derivative"

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" (chapter VIII $\oint 3$, Modular Automorphism groups, p107-108:
Why are the second and third ...

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### If $A$ is a closed operator, is $A^k$ closed?

Let $A$ be a closed (densely defined) operator on a Hilbert space $H$.
We define for a natural number $k$, the operator $A^k$ with its natural domain.
Is $A^k$ closed?

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### Definition of condensed $C^{\ast}$-algebra

The classical definition of a $C^{\ast}$-algebra is a Banach algebra with an isometric antilinear involution map $a \mapsto a^\ast$. What would be a good definition for a condensed $C^{\ast}$-algebra? ...

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### Let $T\in B(H)$. Then prove that $T$ is a contraction if and only if the closed unit disc is spectral set for $T$

Let's first recall the definition of Spectral set for a bounded operator $T$ on a hilbert space $H$. Let $X\subseteq \Bbb{C}$ be compact such that $\sigma(T)\subseteq X$. Define $$\mathcal{R}(X):=\...

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### Extension of $\{f\in C([0, 1], B)\,\vert\, f(0)=f(1)=0\}$ by $A$ with $\ast$-homomorphism $\phi:A\rightarrow B$

The following question is from An Introduction to $K$-theory for $C^{\ast}$-Algebra and an e-copy can be found here. Below is the question (since I do not know how to create a diagram in MS ...)
By ...

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### Identity for spectral resolution: $dE_{\xi, \xi}= |g|^2 dE_{\eta, \eta}$

Let $(\Omega, \mathcal{F})$ be a measurable space. Let $E: \mathcal{F}\to B(H)$ be a regular resolution of the identity on the Hilbert space $H$, see e.g. Rudin's functional analysis book.
Suppose ...

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### What's an easy example of a family of three commuting contractions for which there's no isometric dilation but fulfill the von Neumann inequality?

Or the other way around, an example of a family of three commuting contractions for which there exists an isometric dilation, but which does not satisfy the von Neumann inequality?
The only example I'...

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### Takesaki II Lemma 1.13: stuck in proof

Consider the following fragment from Takesaki's book "Theory of operator algebras II" (Lemma 1.13 on p8, in chapter VI "Left Hilbert algebras"):
Here, we associate with an ...

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### Dilation of bounded linear operators

Let $H$ be a Hilbert space, and let $A$ be a contraction (bounded linear operator of norm $\leq 1$) on $H$. I heard in a recent talk that there is a (apparently famous) result due to Sz-Nagy which ...

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### Let $V = H_{0}^{1}(-1,1) \times H_{0}^{1}$ and $V^{\prime} = H^{-1}(-1,1) \times H^{-1}(-1,1)$ , $T : V \to V^{\prime}$ is compact?

Let $V = H_{0}^{1}(-1,1) \times H_{0}^{1}(-1,1)$ and $V^{\prime} = H^{-1}(-1,1) \times H^{-1}(-1,1)$ the dual space of $V$ and $\lambda \in \mathbb{R}^{\ast}$. Consider $T : V \to V^{\prime}$ ...

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### On a paper of von Neumann

Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality
$$
\lVert p(T)\rVert \leq \sup \...

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### Necessary conditions for $K_0(I_x\bigotimes A)$ to be the trivial group

Let $A$ be a unital $C^*$-Algebra with non-trivial $K_0$ group. Define $CA = \{f\in C\big([0, 1], A\big)\,\vert\,f(0) = 0\}$. It can be shown that $CA$ is homotopic equivalent to the set $\{0\}$ and ...

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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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### Domain of Schrödinger operators

Let $S$ be a Schrödinger operator on $\mathbb{R}$, $Su=-u''+Vu$ with $V\geq1$ continuous and going to $+\infty$ at infinity (you can think of it as $x^2+1$). I wondering which assumptions do I have to ...

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### Algebras sitting inside reproducing kernel Hilbert space other than multiplier algebra

Suppose $\mathcal{H}$ is a reproducing kernel Hilbert space. If the kernel is normalized then the multiplier algebra $\mathcal{M}$ is an algebra that is sitting inside $\mathcal{H}$. Is there any ...

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### Spectral projection with height less than $\lambda$

Let $x\geq 0$ be a positive element in a von Neumann algebra $\mathcal M.$ Then b y functional calculus the projection $e_\lambda=1_{[0.\lambda)}(x)$ has the property that $e_\lambda$ commutes with $x$...

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### Hypo-monotone operators in Hilbert Spaces

If $H$ is a separable Hilbert space, an operator $A \subset H \times H$ is said to be $\lambda$-monotone ($\lambda \in \mathbb{R}$) if
$$ \langle v-w, x-y \rangle \ge \lambda |x-y|^2 \quad \text{ for ...

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### Prove that $N_1(A_1,A_2)= N_2(A_1,A_2)$ when $A_1A_2=A_2A_1$ and $A_1$ and $A_2$ are normal operators

Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$.
On $\mathcal{L}(E)^2$, we have two equivalent norms:
\begin{eqnarray*}
N_1(A_1,A_2)
&=&\sup\...

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### Maximal domain of an unbounded linear operator in a weighted Hilbert-space

Let's consider the following (unbounded) linear operator. (So called transport operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...

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### The $K_0$ mapping of an automorphism induced by a derivation

Let $\mathfrak{A}$ be a unital $C^*$-Algebra and let $\delta: \mathfrak{A} \rightarrow \mathfrak{A}$ be a linear map that is not constantly zero and satisfies, for every $A, B\in\mathfrak{A}$, $\delta(...

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### Spectrum of operator involving ladder operators

The ladder operator in quantum mechanics are the operators
$$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$
and
$$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\...

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### The spectral measure of the integral of an operator-valued function with respect to a projection-valued measure

Let $\mathcal{H}$ be a Hilbert space, $T$ a bounded self-adjoint operator, and $F:\left[a,b\right]\to\mathcal{B}\left(\mathcal{H}\right)$ such that for any $t\in\left[a,b\right]$, $F\left(t\right)$ is ...

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### Sufficient condition for two norms to be equal

Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$.
On $\mathcal{L}(E)^2$, we have two equivalent norms:
\begin{eqnarray*}
N_1(A,B)
&=&\sup\left\...

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### Reference for (general case) of uniqueness of singular value decomposition (SVD)

My statistics research requires me to understand the non-uniqueness of SVD in the degenerate case of repeated singular values.
I believe that the statements and proofs on this StackExchange posts are ...