# 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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### examples of MF algebras [on hold]

Can anyone show me concrete examples of unital MF algebra and non-unital MF algebra respectively? Thanks!

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68 views

### Measurability of the heat semigroup in $L^\infty$

Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$.
It is known that $S(t)$ ...

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33 views

### A question on the Non-degenerated bilinear form [migrated]

Prove that $<C(G), C(G)>$ is dual system with bilinear form $$\int_G \phi(x)\psi(x)dx $$ and $\psi(x),\psi(x)\in C(G)$
so here i was trying bilinear form im not getting how to prove non-...

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62 views

### Existence of a bounded operator which satisfies the discrete product rule

Is there a bounded self-adjoint operator $H$ acting on $\ell^2(\mathbb{Z})$ such that for all sequences $u,v\in \ell^2(\mathbb{Z})$
$$ H(uv)=(Hu)v+u(Hv)$$
where uv is the pointwise product.
This is ...

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86 views

### Polar decomposition of tensor product of operators in von Neumann algebra

If $T=V|T|\text { and } S=W|S|$ is the polar decomposition of $T$. Is it true that the polar decomposition of $T\otimes S$ is $T\otimes S=(V\otimes W)(|T| \otimes |S|)$. If $T$ and $S$ are self-...

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67 views

### power of an infinitesimal generator of a $C_0$ semigroup in a Banach spaces

Let $A$ be the infinitesimal generator of a $C_0$ semigroup of linear operators in a Banach space. Let $n$ pe a positive integer, $n\geq2$. Is $A^n$ closed?
Here (setting $A^1$ $:=$ $A$, and ...

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69 views

### 1D Schrödinger Equation with Measure-Valued Coefficients

I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on $[0,1]$ with Hamiltonian given by the following:
$$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}...

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92 views

### Compact operator

Let $k:[0,1]^2 \to [0,1]$ be a measurable function. Define $K:L^2([0,1])\to L^2([0,1])$ to be the operator:
$$
(Kf)(x) = \int_0^1\int_0^1 f(z) k(x,y) \mathbf{1}_{x\leq z\leq y} \ \mathrm{d}z \mathrm{d}...

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136 views

### Blocksum induces a unital H-space structure on the space of Fredholm operators

Fix a complex separable infinite-dimensional Hilbert space $H$. It is well known that the space of (bounded) Fredholm operators $Fred(H)$ with the norm topology is a classifying space for the ...

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115 views

### A special sequence

I m looking for a sequence $(f_j)\in C^\infty(\Bbb{R})$ such that
$$
\int^\infty_0\Big|\partial^2_r f_j+\frac{1}{r}\partial_r f_j+r^2f_j\Big|^2rdr\to 0,
$$ and
$$\int_{\Bbb{R^+}}|f_j(r)|^2 rdr=1\...

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89 views

### Non-isolated ground state of a Schrödinger operator

Question. Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schrödinger operator $-\...

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56 views

### Spectrum of Dirac sequences

Let $\delta_n\in C^0_c(\mathbb{R})$ be a Dirac sequence approximating the Dirac delta "function" $\delta$ with support in $0\in \mathbb{R}$. Then, for each $n$ we have a compact operator $K_n:L^2(\...

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122 views

### The compact open topology and the operator norm

Let $T:[-1,1]\rightarrow B(H)$ be a continuous family of bounded operators where $B(H)$ is endowed with the compact open topology for an infinite dimesional Hilbert space $H$. Is it true that if $T_0=...

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43 views

### Distributivity of direct sum over maximal tensor product

Let $A,B$ and $C$ be $C^{*}$-algebras.Does the following identity always holds:
$(A \oplus B) \otimes^{max} C \cong (A \otimes^{max}C) \oplus (B \otimes^{max} C)$
My intuition is that this should ...

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143 views

### Bicommutant theorem for commutative operator algebras

Let $\mathcal{B}(H)$ denote the space of bounded linear operators on a complex Hilbert space $H$. The von Neumann bicommutant theorem says:
Theorem. Suppose that $\mathcal{A}$ is a $C^*$-subalgebra ...

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80 views

### On spectral multiplicity of left shift operators

Let $U$ be an operator defined on $l^{2}(\mathbb{Z})$ by $U(e_{n})=e_{n-1}$, where $e_{n}$ is an orthonormal basis of $l^{2}(\mathbb{Z})$. $U$ is a left shift operator. Since $U$ is unitary operator ...

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25 views

### Show that the transition semigroup of the strong solution to a Langevin-type SDE is immediately differentiable

Let
$\varrho\in C^1(\mathbb R)$ with $\varrho>0$
$\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$
$\mu$ denote the measure with density $\varrho$ with respect to $\lambda$
$b:=2^{-...

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161 views

### Graph Laplacian Operator

Consider the linear operator $\mathbb{L} : L^2([0,1])\to L^2([0,1])$ defined by
$$
(\mathbb{L}f)(x) = \int_0^1 xy(f(x)-f(y)) \mathrm{d}y
$$
for all $f\in L^2([0,1])$ and $x \in [0,1]$. Is $\mathbb{L}$...

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66 views

### Trace-class properties of integral operator

Let $k$ be a smooth, compactly supported function defined on $\mathbb{R}^{2}$ and let $Op(k)$ denote the integral operator on $L^{2}(\mathbb{R})$ defined as $$Op(k)f(\cdot)=\int_{\mathbb{R}}k(y,\cdot)...

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33 views

### Stability of the Langevin semigroup under $C_c^\infty(\mathbb R)$

Let
$h\in C^2(\mathbb R)$
$(X^x_t)_{(t,\:x)\in[0,\:\infty)\times\mathbb R}$ be a continuous process on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$X^x_t=x-\frac12\int_0^th'(X^x_s)...

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54 views

### Regularity of superposition operator generated by function between Banach spaces

Let $E$, $F$ be Banach spaces, $D$ be open in $E$, and $K=[0,1]$. Given $\varphi\colon K\times D\to F$ I call
$$
\varphi^\sharp\colon D^K\to F^K,\quad u\mapsto \varphi(\cdot,u(\cdot))
$$
the ...

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59 views

### Gradient bound for the Markov semigroup generated by the solution to an Langevin SDE

Let
$h\in C^2(\mathbb R)$ with $$h''\ge\rho\tag1$$ for some $\rho>0$ and $$\int\underbrace{e^{-h}}_{=:\:\varrho}\:{\rm d}\lambda=1$$
$\mu$ be the measure with density $\varrho$ with respect to the ...

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108 views

### The imaginary exponential of a tangent field on a manifold

If $M$ is a compact Riemannian manifold and $X$ is a tangent field, I am seeking to define the object $\exp {\mathrm i t X}$ for $t \in \mathbb R$, and I do not know how to do it.
One option was to ...

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226 views

### Why is the definition of von Neumann trace independent of the choice of the Hilbert space?

A Hilbert module defined in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück:
A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert space $V$ ...

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77 views

### Singular integral operators and PDEs

What is the relation between the notion of singular integral operators and partial differential equations?
I know, for example, that there is a relation between the Cauchy transform (Riesz transforms ...

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237 views

### Is this consequence of the invariant subspace problem known?

An interesting fact popped out of a paper I'm writing: if the invariant subspace problem for Hilbert space operators has a positive solution, then every $A \in B(\mathcal{H})$ can be made "upper ...

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76 views

### Is the set of points in the irreducible decompositions of this C$^{*}$ -algebra's representations closed?

Suppose $X$ and $Y$ are compact Hausdorff 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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144 views

### construct a concrete $C^*$ algebra

Does there exist a concrete $C^*$ algebra $A$ such that that the following conditions hold:
(1) $A$ is unital and $A$ has no tracial state.
(2)there exists a closed ideal $I$ of $A$ such that $I$ ...

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107 views

### $\sum_{k=1}^dA_k^*A_k$ and $\sum_{k=1}^dA_kA_k^*$ have the same norms if $A_k$ are commuting

Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$.
Let $A_1,\cdots,A_d$ be pairwise commuting operators on $E$. Is the equality
$$\left\|\displaystyle\...

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137 views

### What is the story behind this Hilbert space in the definition of Hilbert Modules

Here is Deflnition 1.5 of Hilbert module in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück:
A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert ...

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108 views

### A single vertex operator - bare bones explanation?

There is an ocean of literature (and a sea of popular texts inside) on vertex algebras, including quite a lot of Q & A here on MO, and I am trying to read some random selections from time to time. ...

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120 views

### Kernel of compact operators [closed]

Since a compact operator $K$ on an infinite dimensional separable Hilbert space $H$ can't be invertible, the spectrum of $K$ must contain zero $0\in \mbox{sp}(K)$. However, $K$ could be injective (e.g....

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211 views

### Stone–von Neumann theorem?

The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...

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75 views

### Interpolation of a trilinear functional

Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If ...

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70 views

### Gradient flows on Hilbert manifolds

I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded.
To be more precise, a ...

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53 views

### equivalent definition of k-rank numerical range of an operator

Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...

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### Inverse Laplacian and convolution in Albeverio's “Solvable Models in quantum mechanics”

I asked this question on math.stackexchange.com two weeks ago but got no answers so far and I got no clues from literature, so maybe someone here knows a reference. I hope it is ok to ask this ...

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65 views

### Some questions on defining the analytic index

The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the ...

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119 views

### A formula related to the Moore-Penrose pseudo-inverse of Hilbert space operators

Let $E$ an infinite dimensional complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$.
Definition: Let $T \in \mathcal{L}(E)$. The Moore-Penrose inverse of ...

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38 views

### Convergence of Eigenvalues and Eigenvectors for Uniformly Form-Bounded Operators

Suppose that $A$ is an operator on a dense domain $D(A)\subset L^2$ with compact resolvent, and with quadratic form $q(f,g):=\langle f,Ag\rangle$.
Let $(r_n)_{n\in\mathbb N}$ be a sequence of ...

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95 views

### a nuclear $C^*$-subalgebra in $\prod_n M_n(\Bbb C)$

Does there exist a non-unital nuclear $C^*$ algebra $A$ of $\prod_nM_n(\Bbb C)$ such that $A$ properly contains $\oplus_n M_n(\Bbb C)$ and each element $(x_n)\not \in \oplus_n M_n(\Bbb C)$ we have $\...

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### Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum

I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...

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72 views

### k-rank numerical range of an operator

Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...

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122 views

### When do supremum and expectation commute?

This is an alternative form of the question in When do maximum and expectation commute?
When we looking for conditions on $G(t,x(t))$ such that
$$
\sup\limits_{t\in [0,N]}E[G(t,X(t))]=E[\sup\limits_{...

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89 views

### Does this element belong to all powers of the augmentation ideal of the group algebra.

Let $G$ be a torsion free group, and let $\alpha$ and $\beta$ are elements in the augmentation ideal, $I$, of $\mathbb CG$, the group algebra of $G$. Assume that there exists complex numbers $a$ and $...

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76 views

### Are these kinds of “crossed product” studied?

Let $M$ be a von Neumann algebra acting in a Hilbert space $H$, and let $\rho$ be a representation of a group $G$ on a Hilbert space $K$. Define $M\rtimes_\rho G$ to be a von Neumann algebra acting in ...

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64 views

### About crossed product of the group von Neumann algebra

Let $G$ be a group with a normal subgroup $N$. If $K$ is a field, then it is well known in ring theory that $K[G]\cong K[N]*(G/N)$ ($*$ stands for the crossed product). Do we have such an isomorphism ...

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69 views

### Convergence of self-adjoint elements in $\sigma$-weak topology

In a von Neumann algebra, if $A_{\alpha}$ converges to $0$ in the $\sigma$-weak topology, do the positive parts $(A_{\alpha})_{+}$ necessarily converge to $0$ in the $\sigma$-weak topology?

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47 views

### Theta Summable operator with bounded trace

Let $D$ be an unboudned self-adjoint operator on the Hilbert space $H$. We assume that all spectrum of $D$ are eigenvalues and $D$ is theta-summable, i.e. $e^{-tD^2}$ is of trace class for all $t>...

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159 views

### Compact images of nowhere dense closed convex sets in a Hilbert space

Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$.
Question. Is there a non-compact linear bounded operator ...