# Questions tagged [operator-theory]

Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.

1,158
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### Norm inequalities for sums of two basic elementary operators

I'm working on this paper. What I'm interested in is this theorem:
where $M_{A,B}(X)=AXB$
I don't know why we can find those two sequences $X_n$ and $x_n$, either I'm finding difficulties to show the ...

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### Takesaki lemma 1.16 (volume II, chapter VII)

I am trying to understand the proof of the implication $(i)\implies (ii)$ in Takesaki's book "Theory of operator algebras II", chapter VII, which says the following:
The relevant setting ...

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### Embedding finite dimensional subspaces of Schatten p-classes

I have recently read a paper by Talagrand: Embedding subspaces of $L_{1}$ into $l^{N}_{1}$, and part of the chapter: Finite dimensional subspaces of $L_{p}$, written by Jonson and Schechtman. The ...

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### Potential scattering for non-decaying potential

I am reading this note on potential scattering by Siu-Hung Tang and Maciej Zworski.
Just wondering if there is a scattering theory for the Schrödinger with a non-decaying potential $- \Delta + V$ on ...

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### On the eigen vectors of a diagonalizable matrix

Let us consider the space $M_n(\mathbb{C})$. By a unitary matrix $U=(u_{ij})$ we mean that $U^{-1}=(\overline{u_{ji}})$.
Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ ...

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### Polar decomposition commute

Let $ X=U|X|$ be the polar decomposition operator with $U$ is unitary operator and $A$ is normal operator and
$AX=XA$
Therefore
$AU|X|=U|U|A$
$A|X|^{1/2} U|X|^{1/2}=|X|^{1/2} U|X|^{1/2} A$
$A\tilde{X} ...

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### Kernels of sequences of operators

Let $\left( S_{n}^{1}\right) $ and $\left( S_{n}^{2}\right) $ two sequences
of operators in $\mathcal{L}(E_{1},F_{1})$ and $\mathcal{L}(E_{2},F_{2})$
where $E_{i},F_{i},i=1,2$ are Hilbert spaces such ...

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### Generalized functional for solution of PDEs

Asked this on Math Stack Exchange awhile ago but it got ignored then deleted.
To solve a differential equation of one variable, you need constraints equal to the number of derivatives.
For a partial ...

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129
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### Discretizing a differential operator which is a function of the derivative operator

Assume that $p(x)$ and $f(x)$ are sufficiently smooth functions and $D\equiv \frac{d}{dx}$. My question is concerned with the discretization of $p(x+D)f(x)$.
As an example, let $p(x)=x^{2}+2x$. Then ...

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### Infinitesimal generator of a Markov process acting on a measure

Short version: The transition operator of a Markov process can act on measures (on the left) or functions (on the right). The infinitesimal generator acts on functions. Is there a way to understand ...

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### Projections in von Neumann algebra tensor product

Let $\mathcal M\subseteq B(\mathcal H)$ be a von Neumann algebra with normal faithful semifinite trace $\tau.$ Consider the von Neumann algebra $\mathcal N:=L_\infty([0,1])\overline{\otimes}\mathcal M$...

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### A trace inequality between self-adjoint operators

Let $A$ and $B$ be self-adjoint operators on some Hilbert space and $B$ is postive. Suppose we have $-B\leq A\leq B$.Is it true then that $\|A\|_p\leq\|B\|_p$ where $\|.\|_p$ is the Schatten-$p$ norm ...

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### Injectivity of a convex combination of squares $A^*A$ in $\ell^\infty$

Consider two operators $A,B: \ell^p \to \ell^p$ ([defined and]bounded for all $p \in [1,\infty]$) as well as their adjoints $A^*,B^*:\ell^p \to \ell^p$. Assume $A^*A$ and $B^*B$ have trivial kernel ...

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### Non-injectivity of the limit of non-injective sequence of operators

It is known that the limit of a sequence of non injective operators is not necessarily non-injective, for instance, the operator \begin{eqnarray*}
T_n &:&\ell ^{2}\rightarrow \ell ^{2} \\
x &...

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### Examples of TRO $V $ and $C^{\ast} $-algebra $B $ for which $V\otimes^hB $ is a TRO

Let $V $ be a ternary ring of operator(TRO) and $B $ be a $\mathbb {C}^{\ast} $-algebra. Let $V \otimes^hB $ denotes the Haagerup tensor product of $V $ and $B $. Obviously if $V $ or $B $ is $\mathbb ...

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91
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### Regarding variation of spectra

I have been reading the article The variation of spectra by J.D Newburgh. in this article and all related reference/ articles, the term 'variation of spectra' keeps coming in, but I nowhere find a ...

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### Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$

My research needs help in finding examples of unitary matrices $U$ which have entries
\begin{align}
U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...

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### Extreme points of the unit ball of bounded operators on $L^p(\mathbb{R}_+)$

Is it known what the set of all extreme points of $B(L^p(\mathbb{R}_+))$ (bounded operators on $L^p(\mathbb{R}_+)$) is?

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### Unicellular compact operators

An operator $T$ on a separable Hilbert space $H$ is called unicellular if any two closed invariant subspaces $M$ and $N$ are comparable; that is either $M\subseteq N$ or $N\subseteq M$. There are many ...

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### Is it true that $\omega = \sum_{(k,l)\in I^2}\omega(p_k - p_l)$

Let $\{p_i\}_{i \in I}$ be a collection of projections in a $C^*$-algebra $A$ such that $\sum_{i \in I} p_i = 1$ in the strict topology (note here that $1$ is the unit of the multiplier algebra $M(A)$)...

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### Structure of the inverse of a Fredholm integral operator of the second kind

NOTE: Cross-posted on Mathematics Stack Exchange
I am trying to solve an equation of the form
$$ (\mathbb{I} + K)\phi = f $$
where $(\mathbb{I} + K): L^2([0,1];\mathbb{R}) \rightarrow L^2([0,1];\...

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### Why is this norm called the maximal norm on the tensor product of ternary rings of operators?

Let $V$ and $W$ be ternary rings of operators (TROs). In section 5 of Kaur and Ruan - Local Properties of Ternary Rings of Operators and Their Linking $C^*$-Algebras, the maximal tensor product $\...

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### Banach-Mazur distance between Schatten-$p$ classes

Let $M_n$ denote the set of all $n\times n$ complex matrices. Let $1\leq p<\infty.$ For $A\in M_n$ define $\|A\|_p:=(Tr(A^*A)^{p/2})^{1/p}$ where $Tr$ denotes the usual trace of a matrix. Then $\|.\...

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### Is it true that every closed ideal $J$ of $ A \otimes_hB$ is of the form $\epsilon^{-1}(I)$ for some closed ideal $I$ of $A \otimes_\text{min} B$

Let $A$ and $B$ be $C^{\ast}$-algebras. It is known that the map $\epsilon:A \otimes_h B \to A \otimes_\text{min} B$ is injective.
Is it true that every closed ideal $J$ of $ A \otimes_hB$ is of the ...

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### Reed-Simon Vol. IV: Question regarding convergence of eigenvalues

I am reading through Chapter XIII.16 of Reed and Simon's Methods of Modern Mathematical Physics IV: Analysis of Operators about Schrödinger operators with periodic potentials. Since the topic is kind ...

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### The closure of selfadjoint elements of an algebra whose spectrum consist of rational numbers

Let $H$ be a seperable complex Hilbert space. What is the closure of the set of all self adjoint operators in $B(H)$ whose spectrum is a subset of the rational number $\mathbb{Q}$.
Apart from finite ...

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### Cyclicity of composition operators

Let $E=C([0,1]^m,\mathbb{R}^n)$ where $K=[0,1]^m$ where $E$ has the compact convergence topology. Recall that for a function $f:[0,1]^m\rightarrow [0,1]^m$ the associated composition operator $C_f$ ...

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### Extended adjoint of Volterra operator

Let $V$ be a Volterra operator on $L^2 [0,1]$.
Does there exist a nonzero operator $X $ satisfying the following system
$VX=XV^∗$, where $V^∗$ is the adjoint of the Volterra operator?
$$ V(f) (x) =\...

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### Estimates of the Frobenius norm of commutator

Let $A,B$ be two unitary matrices in $U(n)$, and $\|\cdot\|_{F}$ denote the Frobenius norm (or Hilbert Schmidt norm on the finite dimensional $M_n(\mathbb{C})$). I am looking for estimates of the ...

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### Norm of operators similar to a unitary one

Let $U$ be a unitary operator on a Hilbert space $H$ and $Q \in B(H)$ is positive and invertible, is it true that if $\|QUQ^{-1}\| = 1$, then $Q$ commutes with $U$?
One can show that this is the case ...

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### Exponentials and other functions of sums of anti-commuting operators

I know that if $A$ and $B$ are commuting operators, then $\exp(A+B) = \exp(A) \exp(B)$. Is there a similar formula if $A$ and $B$ are anti-commuting (that is, $AB+BA = 0$)?
I have developed a formula ...

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### Need reference for: $\lVert\cdot\rVert_{\text{max}} \leq \lVert\cdot\rVert_h$

Let $A$ and $B$ denote $C^{\ast}$-algebras. Let $\lVert\cdot\rVert_h$ and $\lVert\cdot\rVert_{\text {max}}$ denote the Haagerup norm and max $C^*$-norms on $ A \otimes B$, respectively. I am looking ...

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### The density of the image of a unitary irrep (a generalization of Burnside's theorem)

I asked the following question on MSE and never got an answer.
I am curious if there are any generalizations of Burnside's theorem (If $(\pi,V)$ is irreducible, then $\pi(G)$ spans $\operatorname{End}(...

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### $C_{0}$-semigroups with growth bound zero: how do constants $M_{\omega}$ grow as $\omega\to 0^{+}$?

Let $\{T(t)\}_{t\geq 0}$ be a $C_{0}$-semigroup on a Banach space $X$ and define the growth bound to be the quantity
\begin{equation} \omega_{A}=\inf\{\omega\in\mathbb{R} \mid \exists\text{ } M_{\...

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### Closure of finite rank operators on $L^p$

It well-known that, an operator $T:H\to H$ on a Hilbert space, is compact if and only if T is limit of finite rank operators.
Besides this, the results by Per Enflo 1973 shows that this results is ...

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### "Cyclic vector" of sequence of operators

I recently encountered the following somewhat random-looking problem in my research. At first I thought that should not be too hard, but now, the more I think about it, the more interesting it seems.
...

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### Antilinear unbounded operator has closed graph

Let $H$ and $K$ be Hilbert spaces and $D(T)$ a vector subspace of $H$. Let $T: D(T) \to K$ be a densely defined antilinear operator. Its adjoint $T^*: D(T^*)\to K$ is defined by the relation
$$\langle ...

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### Let $V$ be a TRO such that $A(V)= \mathbb{C}$, what can we say about $V$?

Let $V$ be a TRO i.e. closed subspace of $B(H,K)$ such that $xy^*z \in V$ for all $x,y,z \in V$. Let $C(V)$ and $D(V)$ denotes the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^*V$ respectively. ...

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### Bounded linear operator on a normed space with bounded inverse and dense range

Does there exist such an operator $T$ (bounded and 1-1 on a normed space $X$, it's range $R_{T}$ is dense in $X$ and $T^{-1}: R_{T}\to X$ is bounded) which is not surjective? In other words, does ...

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### On which subspace $W\subset C^{\infty}[0,1]$ is $(Df)(x)=xf'(x)$ a bounded operator provided all functions in $W$ are flat functions?

I have already asked this question on MSE; now I repeat it on MO.
https://math.stackexchange.com/questions/4132346/on-which-subspace-w-subset-c-infty0-1-is-df-xfx-a-bounded-operator
First we ...

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### Reference request for spectral theory of elliptic operators [closed]

I want to learn the spectral theory of linear elliptic operators in bounded and unbounded domains in $R^n$, in particular for Laplacian and Schrodinger operators. Please suggest me some reference.
I ...

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### Example of a unital contractive map that is not completely positive on an operator system

I am aware of maps that are positive but not completely positive (for example transpose map). BUT I can not think of an example of the following type.
Does there exist an operator $T$ such that a map $...

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285
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### Relaxation of notion of positive definite function

A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...

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### Takesaki: Lemma about enveloping von Neumann algebra

Consider the following lemma with proof from Takesaki's book "Theory of operator algebra I" (p121):
It appears to me that Takesaki claims at the end of the proof that $\pi(A)_1$ is $\sigma$-...

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### Takesaki's proof of the Kaplansky density theorem

Consider the following fragment from Takesaki's book "Theory of operator algebra I":
Why is the boxed sentence true? It looks like they replace $A$ by its strong$^*$-closure. Is this ...

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### Stinespring's theorem: can we choose the dilation to be an isometry?

Let $A$ be a $C^*$-algebra and $\varphi: A \to B(H)$ be a completely positive contractive map. Stinespring's theorem says that there exists a $*$-representation $\pi: A \to B(H')$ and a bounded ...

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112
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### A question on the self-adjointness of an operator

Given a Hilbert space (separable) $\mathcal{H}$ with an orthonormal basis $\{e_i\}_{i=1}^{\infty}$, define an operator $T$ with domain $\mathcal{D}(T)$ equal to the span of $\{e_i\}$ by $Te_i:=\...

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### Norm antipode on a Kac-type compact quantum group

Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group. Consider the associated dense Hopf$^*$-subalgebra $\mathcal{O}(\mathbb{G})$ and let $S: \mathcal{O}(\mathbb{G})\to \mathcal{O}(\mathbb{G})$ ...

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### Largest nuclear norm of $n \times n$ symmetric matrices whose entries are between -1 and 1

Let $\cal M$ be the set of real symmetric $n \times n$ matrices whose entries are all in the interval $[-1, 1]$. I'm interested in understanding the largest possible nuclear norm of these matrices as ...

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### Criteria for operators to have infinitely many eigenvalues

Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem.
For non-normal operators this no longer has to be true.
There ...