# 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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### An example of non trivial projections in a group von Neumann algebra

Let $G$ and $\text{vN}(G)$ be a torsion free group and its group von Neumann algebra. Is there a characterization of non trivial projections in $\text{vN}(G)$? If not, is a certain class of them ...

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147 views

### An analytical zero divisor

Let $G$, $\mathbb C[G]$ and $\text{vN}(G)$ be a torsion free group, it's group ring and group von Neumann algebra, resp.. Let $0\neq\alpha\in\mathbb C[G]$ and $0\neq p\neq1$ is a projection in the ...

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69 views

### Superposition operator from Sobolev space to Lebesgue space

Let $\Omega$ be a bounded, connected set in $\mathbb{R}^2$ with smooth boundary. I am wondering under what conditions on the real function $f(x):\mathbb{R}\to \mathbb{R}$ the superposition operator $F(...

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126 views

### Compact embedding result

Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...

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82 views

### Fredholmness of formal selfadjoint operator $AA^*$ and Fredholmenss of $A$

Let $X$ and $Y$ be Hilbert spaces with respective inner products $\langle , \rangle_{X,Y}$. Let $A:X \rightarrow Y$ be a bounded linear operator. Assume there is a non-degenerate sesquilinear product $...

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57 views

### Disreteness of spectra of Dirichlet laplacians

I have fundamental questions on Dirichlet Laplacians.
Let $D \subset \mathbb{R}^d$ be an open subset and $\mathcal{L}$ be the (non positive) Dirichlet laplacian on $D$.
We denote by $T_t=e^{t\...

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115 views

### Schrödinger operator with Coulomb potential

The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\...

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109 views

### Energy in doubling measure metric spaces

Let $(X,\mu, d)$ be a metric measure space where $\mu$ is a doubling measure. For a relatively compact set $U\in X$ consider the following quantity
$$I(U,\mu,d)=\int_U \int_U \log^2(d(x,y)) d\mu(x) d\...

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**1**answer

275 views

### Criterion for a Banach algebra to be finite dimensional

Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional.
Question. Does it follow that $A$ is finite dimensional?
...

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302 views

### Does point-wise weak convergence give weak convergence in $L^2(I;X)$?

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, ...

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72 views

### Common eigenvector of commuting unbounded Operators

Let be T,S two self adjoint linear Operator on a Hilbert Space $\mathcal{H}$ with pure point spectrum.
Then T,S commute if and only if they have a complete set of common eigenvectors.
This is the ...

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165 views

### Is the kernel of a Fredholm operator stable under perturbation?

This is a follow-up of this question.
In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator?
Let $(X,\| \cdot \|)$ be an infinite-dimensional normed vector space.
...

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82 views

### Conditional Expectation for von Neumann algebra

Let $M$ be a von Neumann algebra and $T: M\rightarrow \mathbb{C}$ be a finite normal faithful tracial map, s.t., if $\phi : M \rightarrow A \cap A^{*}$ is a conditional expectation, (A being a weak ...

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214 views

### Elementary quantum scattering problem on the line.

Let us consider the quantum scattering problem on the line with the Hamiltonian
$$H=-\frac{d^2}{dx^2}+ V(x),$$
where $V(x)=1$ when $x\in (0,a)$, and $V(x)=0$ otherwise.
It is easy to see that $H$ ...

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**1**answer

192 views

### Can this self-adjoint operator have an infinite-dimensional compression with compact inverse?

The following might be quite straightforward, but I very rarely work in detail with unbounded operators, so I thought it would be worth seeing quickly if I have overlooked an example that is obvious ...

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109 views

### Closure of tensor product /tensor product semigroup

In this reference the following claim is made in Remark 2
Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...

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33 views

### Invariance of simple functions

Let $(A(s),D(A(s)))_{s\in\mathbb{R}}$ be a family of unbounded operators on a Banach space $X$ and $g:\mathbb{R}\rightarrow X$ be a simple function, i.e.,
\begin{align*}
g=\sum_{i=1}^n{x_i\textbf{1}_{...

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298 views

### Hölder continuity for operators

Let $x,y$ be positive real numbers then
$$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$
we obtain $1/...

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39 views

### Is singular Cauchy operator bounded in Morrey spaces?

The singular Cauchy operator is defined by
$$S_\Gamma :f \to \int_\Gamma \frac{f(\xi)}{\xi-z} d\xi , z\in \Gamma.$$
Is this operator bounded in Morrey spaces and weighted Morrey spaces? i.e.
is there ...

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**1**answer

336 views

### One question about the $\eta$ invariant

This question is from the paper, The Analysis of Elliptic Families
II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8.
Suppose ...

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127 views

### The tensor product of two bounded operators

Let $E$, $F$ be two complex Hilbert spaces and $\mathcal{L}(E)$ (resp. $\mathcal{L}(F)$) be the algebra of all bounded linear operators on $E$ (resp. $F$).
The algebraic tensor product of $E$ and $F$ ...

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181 views

### Absence of fixed points

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed ...

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97 views

### Compactness of operators of Markov processes

I have a question about operators of Markov processes.
Let $(E,\mu)$ be a locally compact separable metric measure space. Let $X=(X_t,P_x)$ be a $\mu$-symmetric Markov process on $E$. For any bounded ...

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249 views

### Function square-integrable

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$
where $x_0$ is an ...

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268 views

### Uniqueness of solution depending on constant?

I am a physicist and I am aware that this forum is for professional mathematical questions, but please be not too hard on my notation.
I encountered the following integral equation for functions $f:[...

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100 views

### Closeness of points in the irreducible decomposition of a C$^{*}$-algebra representation

Suppose $X$ and $Y$ are compact metric spaces. Let $\varphi\colon C(X)\to M_{n}(C(Y))$ be any $*$-homomorphism. If $\pi$ is an irreducible representation of $M_{n}(C(Y))$, then $\pi$ is unitarily ...

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55 views

### Relative boundedness of the adjoint

Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$
...

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130 views

### Description of (completely) bounded operator

I am somewhat a beginner in the field of operator algebras and was wondering about the following:
Let $T$ be a linear map between the space of bounded operators $B(H)$ on some Hilbert space and $S$ a ...

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70 views

### Difference Between Eigenvalues of Schrödinger Operator with Different Boundary Conditions

Consider a Schrödinger operator
$$H=-\Delta+V$$
on a nice bounded domain $\Omega\subset\mathbb R^d$ (say, a ball or a cube), and assume for simplicity that $V$ is smooth.
Let $\lambda_D,\lambda_N$ ...

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65 views

### Propagation of singularities and the Schrodinger equation

I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...

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116 views

### Invariant subspace in infinite dimensions

Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$
The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...

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148 views

### $L^2$ function in Schwartz space?

Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$
Such a function has the property that when multiplied with any ...

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100 views

### Nuclear operators and their eigenvalues

I know that if an operator (on a Banach space with approximation property) is nuclear of order zero, then its eigenvalues are $p$-summable for any $p>0$. (I read it from Grothendieck’s book “...

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68 views

### What is the generator of the heat semigroup on non-complete manifolds?

If $M$ is a complete Riemannian manifold, it possesses a unique self-adjoint positive operator $-\Delta$ on $L^2(M)$. If $M$ is not complete, though, it is known that the Laplace-Beltrami operator $-\...

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42 views

### Uniform convergence in Hadamard derivatives

Let $T\colon X \to Y$ be a nonlinear operator between Hilbert spaces which is Lipschitz and is Hadamard differentiable. It satisfies
$$T(x+th)=T(x) + tT'(x)(h) + r(t)$$
where $r(t)=r(t,x,h)$ is the ...

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72 views

### norm of operator between matrix algebras equipped with trace norm [duplicate]

Let $M_i$ stands
for the algebra of $d_i\times d_i$ matrices with $\|T\|=d_i
\|T\|_1=d_i (trace{(T^\ast T)}^{\frac{1}{2}})$, and $M_{ij}$
stands for the algebra of $d_i d_j\times d_i d_j$ ...

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76 views

### Analytic families of compact self-adjoint operators: eigenvalue extension

Suppose that $A(t), t \in \mathbb{R}$, is an analytic family of compact self-adjoint operators on a Hilbert space. The Kato-Rellich theorem says that every non-zero eigenvalue of $A(t)$ splits into ...

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120 views

### Is this a pseudodifferential operator?

Let $M$ be a non-compact manifold and $D$ a first-order self-adjoint elliptic differential operator on $M$. Then is the bounded operator
$$A:=\sqrt{(D^2+1)^{-1}}:L^2(M)\rightarrow H^1(M)$$
a ...

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139 views

### Does separability of the strong operator topology imply separability of the underlying space?

Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$.
Suppose that the strong operator topology on $B(X)$ is separable and that the cardinal number of $B(X)$ is continuum.
...

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286 views

### If the cardinality of $B(X)$, the space of operators on $X$, is continuum, must $X$ be separable?

Does there exits any non-separable Banach space $X$ such that the size (cardinal number) of $B(X)$, bdd linear operators on $X$, is just of the continuum?

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297 views

### Approximate eigenvectors for a set of non-commuting self-adjoint operators

This problem is motivated by finding the right mathematical setting for expressing the compatibility of classical physics with quantum mechanics.
Let $\mathcal H$ be a Hilbert space and $S$ a ...

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70 views

### Convergence rate of eigenvectors

Let us suppose that $A,A_1,A_2,\ldots$ are non-negative definite self-adjoint bounded linear operators in $L(\mathbb H)$, where $\mathbb H$ is a separable Hilbert space. $(v_j)_{j\ge1}$ and $(\...

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133 views

### References for the Spectral Theorem ( Multiplication Operator Form)

Let $A_1$ and $A_2$ be two commuting self-adjoint (or normal) operators on an infinite-dimensional complex Hilbert space $E$, then there exists a measure space $(X,\mathcal{E},\mu)$,
two functions $\...

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67 views

### Spectral projection of a positive operator

Let $(X,K)$ be a partially ordered Banach space where the cone $K$ is generating and normal. Suppose $B$ is a bounded operator on $X$ such that $B(K)\subseteq K$, the spectral radius $\rho(B)=1$ and ...

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74 views

### Are the Prolate Spheroidal Wave Functions absolutely integrable?

I would like to know if the Prolate Spheroidal Wavefunctions (PSWFs, defined below) are in $L^1(\mathbb{R})$. I know that they are square integrable, but cannot decide about absolute integrability.
...

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72 views

### Spectrum of a Hamiltonian on the real line

Consider the following linear (Hamiltonian) operator on functions on the real line $\mathbb{R}$
$$H\psi(x)=-\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x).$$
Assume that $V$ is a smooth function and $V(x)\to +\...

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147 views

### Bound on sum of $n$th super-diagonal entries in a $2n$ by $2n$ PSD matrix

Let $A,B,C\in\mathbb{R}^{n\times n}$ be such that
$\left(\begin{array}{} A & B \\ B^T & C \end{array}\right)\succeq 0$. I would like to prove that
$$\mathrm{trace}\,B \le \sum_{i=1}^n \sqrt{\...

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136 views

### The nonlinear operator defined as the commutator of a matrix and a nonlinear operator

In my studies of applied analysis and applied linear algebra, this interesting problem and concept came up:
Let us consider the space of all $ m \times n $ real matrices, and define a scalar ...

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106 views

### The norm of a bounded operator acting on $\operatorname{Im}(M^{1/2})$

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space $(F,\langle\cdot,\cdot\rangle)$. Let $M\in \mathcal{B}(F)^+$ (i.e. $\langle Mx\;, \;x\...

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135 views

### On projection theory for inseparable Hilbert spaces

How can one see that $I$ is an infinite projection in $B(\mathcal{H})$, where $\mathcal{H}$ is an inseparable Hilbert space?