# Questions tagged [operator-theory]

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

416
questions with no upvoted or accepted answers

**46**

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### Set-theoretic reformulation of the invariant subspace problem

The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$...

**29**

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2k views

### Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $||x|| = 1$ and $||Ax|| = ||A||$. The ...

**27**

votes

**0**answers

1k views

### Unital $C^{*}$ algebras whose all elements have path connected spectrum

A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...

**18**

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**0**answers

154 views

### Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$

Remark: I cross-posted this question on MSE and added a bounty to it.
Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...

**17**

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310 views

### Tauberian Theorem for 1-parameter groups of operators

The Wiener Tauberian Theorem gives condition on an $f\in L^1(\mathbb{R})$ such that the "induced 1-parameter family" $\{T_b(f)\}_{b\in \mathbb{R}}$ has a dense span in $L^1(\mathbb{R})$; ...

**15**

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**0**answers

2k views

### PT Symmetry and the Riemann Hypothesis

Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The ...

**11**

votes

**0**answers

461 views

### Contraction semigroup on Hilbert space

I'd like to know whether a certain unbounded operator on a Hilbert space is the generator of a strongly continuous contraction semigroup.
(Such operators are known as maximally dissipative operators.)
...

**10**

votes

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195 views

### Norm-attaining operators with values in a 2-dimensional Hilbert space

Is the set $N\!A(X,\ell_2^2)$ of norm-attaining operators from a Banach space $X$ onto the $2$-dimensional Hilbert space $\ell^2_2$ dense in the Banach space $L(X,\ell_2^2)$ of all linear continuous ...

**10**

votes

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231 views

### Do sufficiently large Banach spaces admit non-compact operators with not too large range?

As in the title,
does there exist a cardinal number $\lambda$ such that for every Banach space $X$ of density/cardinality at least $\lambda$ there exists a non-compact bounded, linear operator $T\...

**10**

votes

**0**answers

100 views

### Holomorphic natural bundles and operators

I am wondering up to what extent the classical theory of (smooth) natural bundles and natural operations extends to the holomorphic setting.
After a quick thought, I've gone through the standard ...

**10**

votes

**0**answers

633 views

### Functional calculus of unitary matrices and commutator norms: reference request

Suppose we have a normal matrix $A$ and a general matrix $B$.
For a continuous function $f$ on the disk we can find upper-bounds
on $\Vert[f(A),B]\Vert$
in terms of $\Vert[A,B]\Vert$. The more we know
...

**9**

votes

**0**answers

213 views

### Von Neumann Inequality in Banach spaces

It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space:
Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...

**9**

votes

**0**answers

130 views

### Comparison of the absolute value of an operator with its positive parts, II

Suppose $A,B\in M_n(\mathbb C)$ are self-adjoint. Does there exist a constant $C>0$ depending only on $n$ such that
$$
|A+iB| \leq C(|A| + |B|)?
$$
One can take $C=1$ if $A$ and $B$ commute.
More ...

**8**

votes

**1**answer

214 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" ...

**8**

votes

**0**answers

244 views

### Struggling with a proof in the preprint 'Hermitian geometry on resolvent set'

I have been struggling for awhile with a particular argument in the paper below. I posted the question first on MathSE, but I got no answers. I understand however that MO might be an overreach for ...

**8**

votes

**0**answers

203 views

### Shift on trivalent directed tree, operator and von Neumann algebra

Let $\mathcal{T}$ be the trivalent directed tree, with two parents and one child for each vertex (see below). Let $\mathcal{V}$ be the set of vertices of $\mathcal{T}$ and $H$ be the Hilbert space $\...

**8**

votes

**0**answers

236 views

### Lax pairs in an abstract formalism

I am reading Integrals of Nonlinear Equations of
Evolution and Solitary Waves by Peter Lax and I'm having a hard time. The methods are pioneering, of course, but Lax does not bother much to provide ...

**7**

votes

**0**answers

182 views

### Monotonicity of log determinant of Gaussian kernel matrix

Let \begin{equation} k({x},{y}) = \sigma \exp\left(-\frac{(x-y)^2}{2\theta^2}\right)\end{equation}
be a squared-exponential (Gaussian) kernel, with $\sigma,\vartheta>0$. Consider, for a set of $N$ ...

**7**

votes

**0**answers

72 views

### Analogue of Friedrichs extension for Hilbert $C^*$-modules

Suppose one has a densely defined symmetric operator $T:\mathcal{M}\rightarrow\mathcal{M}$, where $\mathcal{M}$ is a Hilbert $A$-module for a $C^*$-algebra $A$. Suppose that $T$ is non-negative, so ...

**7**

votes

**0**answers

164 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. ...

**7**

votes

**0**answers

348 views

### Spectra of Dirac operators

1. Suppose that $M$ is a spin manifold. The spin structure of $M$ is not uniquely defined: in other words, $M$ may have many
nonequivalent spin structures. For each choice of spin structure there is ...

**7**

votes

**0**answers

171 views

### Some questions on the nodal geometry of Dirac operators

Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows:
Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...

**7**

votes

**0**answers

283 views

### Fock Space Proof of $(g(x)\phi^4)_2$ Mass Gap?

Is there a proof that does not depend on Euclidean methods? Is this a proof? :
$V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). $...

**7**

votes

**0**answers

161 views

### Is there a quotient of $c_0$ without the approximation property?

The famous example of Enflo of a Banach space without the approximation property is actually a subspace of $c_0$. Is there a quotient of $c_0$ without the approximation property?
This would follow if ...

**6**

votes

**0**answers

183 views

### For what kind of $C^*$ algebra $A$ every normal element $y\in A$ has a normal lift for every given surjective $C^*$ morphism $\phi:B\to A$

Is there a terminology for the following property of $C^*$ algebra $A$:
For every $C^*$ algebra $B$ and surjective $C^*$ morphism $\phi: B\to A$, every normal element $y\in A$ admits a normal ...

**6**

votes

**0**answers

94 views

### Understanding the odd-dimensional index

Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...

**6**

votes

**0**answers

150 views

### Question on operator topologies convergence

Let $H$ be a complex Hilbert space, and let $\mathcal{B}(H)$ denote the algebra of bounded operators on $H$. It is known that the strong operator topology and the norm topology on $\mathcal{B}(H)$ ...

**6**

votes

**0**answers

164 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 ...

**6**

votes

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139 views

### $L^p$ estimates for solutions of strongly elliptic equations

Let $\Omega\subset \mathbb{R}^d$ be a Lipschitz bounded domain. In $L_2(\Omega;{\mathbb C}^n)$, we consider a strongly elliptic second order operator $A$ with constant coefficients. Namely, it is ...

**6**

votes

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183 views

### Generalized singular numbers and the Haagerup $L^p$ spaces

Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...

**6**

votes

**0**answers

497 views

### Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term

Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...

**6**

votes

**0**answers

57 views

### Chord-arc property of n-tuples of commuting operators

Assume that we have an $n$-tuple $S^0=(S^0_1,\dots,S^0_n)$ of commuting operators in a Hilbert space $H$ and another such an $n$-tuple $S^1$. Is it possible to connect these two $n$-tuples by a ...

**6**

votes

**0**answers

224 views

### Operator arithmetic-harmonic mean inequality with operator-valued weights

Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...

**6**

votes

**0**answers

236 views

### How to take this Grassmann integral?

I'm trying to reconstruct and understand what is explained in a paragraph of this paper. I am trying to check if the method they describe actually gives us the Laughlin state. The integral I'm facing ...

**6**

votes

**0**answers

776 views

### About generator and isomorphism problems for free groups operator algebras

Let $H$ be an infinite dimensional separable Hilbert space.
The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$.
Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a ...

**6**

votes

**0**answers

251 views

### What is the intersection of the closures of left invertible operators and right invertible operators?

From Douglas Zare's answer (see Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?), one know that
$$ \overline{G_{l}(X,Y)} \bigcap \overline{G_{r}(X,Y) } = \...

**5**

votes

**0**answers

105 views

### Commutator of pseudodifferential operator and multiplication operator

Cross-post from math.sx.
Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the ...

**5**

votes

**0**answers

95 views

### Which operations commute with fractional translation?

Let $\mathbf{v}$ be a real signal (i.e., an infinitely long vector).
A translation operation $T_{s}\mathbf{v}$ with integer $s$ is trivially
defined by $\forall i,s:\left(T_{s}\mathbf{v}\right)_{i}=v_{...

**5**

votes

**0**answers

132 views

### Standard function spaces with the approximation property

A Banach space $\mathcal{X}$ is said to have the approximation property (AP) if, for every compact set $K \subset \mathcal{X}$, there is a sequence of finite rank operators $\{U_n : \mathcal{X} \to \...

**5**

votes

**0**answers

125 views

### Cyclic vectors for the translation operator

Let $b\in \mathbb{R}\neq 0$, and consider the translation operators:
$$
\begin{align}
T_b:C(\mathbb{R}) & \rightarrow C(\mathbb{R})\\
f &\mapsto f(\cdot + b).
\end{align}
$$
*Are there known ...

**5**

votes

**0**answers

108 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 ...

**5**

votes

**0**answers

157 views

### Wightman reconstruction theorem-details of the proof

First of all forgive me if this question is not well suited for this forum: it is motivated by physics however after all my concerns are mathematical so I hope it would be appropriate to post it here. ...

**5**

votes

**0**answers

62 views

### Reference request: a survey of (linear) Krein-Rutman theory

I'm looking for a survey article or book chapter where a rather exhaustive treatment of the Krein-Rutman theory of positive linear operators an ordered Banach spaces is given.
Motivation. Some ...

**5**

votes

**0**answers

173 views

### Index of the Fredholm operator

I have two vector bundles $E_1$, $E_2$ over $M$ and an embedding of the smooth sections $\lambda : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1 \oplus E_2)$. I consider a Fredholm differential operator $...

**5**

votes

**0**answers

127 views

### Copies of $\ell_\infty^k$ in subspaces of the space of operators between $n$-dimensional Banach spaces

Are there a positive integer $k$ and an unbounded increasing function $d:\mathbb N\to\mathbb N$ (of growth order $\Omega(n^2)$) such that for any $n$-dimensional Banach spaces $X,Y$, the Banach space $...

**5**

votes

**0**answers

371 views

### The Riemann zeta function and differential operators

I've revisited an old post of mine--Dirac's Delta Functions and Riemann's Jump Function J(x) for the Primes--dealing with Riemann's "jump" or "staircase" function (aka, Π(x)) that ...

**5**

votes

**0**answers

200 views

### Constructing a noncommutative algebra from a commutative algebra

I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...

**5**

votes

**0**answers

125 views

### Banach space properties defined by compact operators, strictly singular operators and strictly cosingular operators

Let $X,Y$ be Banach spaces. We denote by $\mathcal{L}(X,Y)$ the space of all operators from $X$ into $Y$, $\mathcal{K}(X,Y)$ by the space of all the compact operators from $X$ into $Y$, $S(X,Y)$ by ...

**5**

votes

**0**answers

95 views

### Characterizing Herz-Schur multipliers using coefficient functions of uniformly bounded representations

Let $G$ be a group and let $c > 1$ be a constant. We denote by $B_c(G)$ the space of all coefficients of the representations of $G$ which are uniformly bounded by $c$; more precisely, a function $f:...

**5**

votes

**0**answers

167 views

### Representations of the algebra of shift-invariant operators on $\ell^\infty({\mathbb Z})$

$\newcommand{\Z}{\mathbb Z}$
By an operator on $\ell^\infty(\Z)$, I mean a bounded linear map $\ell^\infty(\Z)\to\ell^\infty(\Z)$. (Note that I am not assuming weak-star continuity.) By shift-...