**36**

votes

**0**answers

1k views

### 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 ...

**23**

votes

**0**answers

497 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 ...

**14**

votes

**0**answers

452 views

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

A unital $C^{*}$ algebra is called "Path connected" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
Is the tensor product of two path connected algebra, a path ...

**10**

votes

**0**answers

428 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
...

**8**

votes

**0**answers

73 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 ...

**6**

votes

**0**answers

131 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

189 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

107 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

242 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

83 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 ...

**5**

votes

**0**answers

100 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 $S$ ...

**5**

votes

**0**answers

198 views

### Eigenvalues of a certain product of matrices with special structure

Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the $q$-by-$2q$ matrix $A = ...

**5**

votes

**0**answers

109 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). ...

**5**

votes

**0**answers

122 views

### When is an inner derivation a Fredholm operator?

Let $\mathcal{B}(H)$ denote the algebra of bounded operators on a Hilbert space $H$. I'm interested in inner derivations acting on the Schatten ideals $L^p\subseteq\mathcal{B}(H)$ (defined by ...

**5**

votes

**0**answers

481 views

### Do the banded operators check the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...

**4**

votes

**0**answers

40 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 ...

**4**

votes

**0**answers

111 views

### Smooth perturbation of a positive self-adjoint operator with compact resolvent

Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian ...

**4**

votes

**0**answers

89 views

### Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, ...

**4**

votes

**0**answers

118 views

### Infinitesimal Generator of Billiard Flow

The Billiard flow $S_t$ on a Riemannian manifold with boundary (with corners) is the group of operators defined on continuous functions on the Co-sphere bundle as follows: To determine $S_t u(\xi)$, ...

**4**

votes

**0**answers

135 views

### A question on extension of $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor.
Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence ...

**4**

votes

**0**answers

145 views

### The representation-theoretic nature of an operator resolvent

Consider parameter $s$ in definition of $R(s,A)=(s I - A)^{-1}$ where $A$ is a linear operator in a vector space $X$. When $X$ is over $\mathbb{C}$, then $s$ is thought to be a complex number.
Now ...

**4**

votes

**0**answers

371 views

### Denseness of finite rank operators in $\mathcal{B}(X,Y)$

Let $X$ and $Y$ be Banach spaces and let $\mathcal{B}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. As noted in the answers to a question on
...

**4**

votes

**0**answers

195 views

### Adjoint of sum of two operators. Kato-Rellich

Let $A$ be self-adjoint and $B$ be symmetric with $A$-bound less than $1$. By Kato-Rellich, I know that $(A+B)^*=A+B$. Could I also get something like $(A+iB)^*=A-iB$ or is there a counterexample to ...

**4**

votes

**0**answers

122 views

### Noncommutative Poincaré inequalities

This is a question on how (or if) people in the community think about the Poincaré inequality in noncommutative geometry. In geometry, the Poincaré inequality (when it exists) gives a bound on a ...

**4**

votes

**0**answers

75 views

### Strictly convex renormings making power bounded operators into contractions

Let $X$ be a Banach space and let $T$ be a power bounded linear operator on $X$ (i.e. $\sup_{n\ge0}\|T^n\|_{op}<\infty$). We can of course define an equivalent norm $\|\cdot\|'$ on $X$ so that ...

**4**

votes

**0**answers

211 views

### Essential unitary equivalence

Let us agree that heuristic meaning of the word "essential" is: up to compact operator. There is clear notion of unitary equivalent operators. What is the proper notion of two operators being ...

**4**

votes

**0**answers

461 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 ...

**4**

votes

**0**answers

117 views

### Containment of an element to an operator system

This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...

**4**

votes

**0**answers

639 views

### The spectrum of a Markov Operator and Invariant Measures

Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...

**3**

votes

**0**answers

71 views

### Spectral theory of Bochner integral operators

Consider the following (somewhat simplified) situation. Let $\mathcal{H}$ be separable Hilbert space and $\mathcal{B}(\mathcal{H})$ the Banach algebra of bounded linear operators acting on ...

**3**

votes

**0**answers

229 views

### Graded structures for simple $C^{*}$ algebras without nontrivial idempotent

Edit(A confession): I just realized that the question is trivial: Since one can easily prove that the convex hull of the spectrum of every nontrivial homogeneous element of a $\mathbb{Z}_{n}$-graded ...

**3**

votes

**0**answers

41 views

### Reference request: derivative of trace of heat operators with respect to a parameter

I hope that the following question is appropriate to ask here as it is not exactly research or original Mathematics but rather an enquiry for a reference or "standard method of proof":
Suppose we are ...

**3**

votes

**0**answers

114 views

### Stability of a linear system and spectrum of the product of two matrices

Let us consider an invertible matrix $\mathbf{A}\in GL_d(\mathbb{R})$ such that all its diagonal entries $\mathbf{A}_{ii}=-1 \; \forall \, i$.
My question is the following:
Does it always exists a ...

**3**

votes

**0**answers

210 views

### The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$

The Dunkl intertwinig operator $V_k$ on $C(\mathbb{R}^d)$ is defined by:
$$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$
where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...

**2**

votes

**0**answers

17 views

### Multivariable Weighted shift and subnormality

I have asked this question in mathstackexchange but didn't get any answer. I hope, I will get my answer here.
Let $\mathbb B^m$ denote the Euclidean ball in $\mathbb C^m.$ Does there exist a ...

**2**

votes

**0**answers

28 views

### When is a linear operator on $C^{0,\alpha}(\overline{\Omega})$ a multiplication?

The title says it all, really. Suppose that I have a linear operator $T$ from $C^{0,\alpha}(\bar{\Omega})$ into itself, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^d$ (e.g. the unit ball ...

**2**

votes

**0**answers

129 views

### The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras

Is there a name for the following property of a $C^{*}$ algebra $A$?
$$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$
Example of this situation is $A=C(X)$ where $X$ is the ...

**2**

votes

**0**answers

133 views

### Infinitesimal generator and stationarity

The following question is bothering me. I think it is probably known but I cannot find any reference...
Let $(X_t)$, $(Y_t)$, $(Z_t)$ denote 3 Feller processes with respective infinitesimal ...

**2**

votes

**0**answers

85 views

### Are Ritt operators mean ergodic?

In the following, $T$ is a bounded operator on a Banach space $X$.
$T$ is called "power bounded" if $\sup_{n\in \mathbb N}\|T^n\|<\infty$;
$T$ is called "mean ergodic" if the Cesàro sums ...

**2**

votes

**0**answers

30 views

### Specific type operators and basic sequences

Let $s$ be the space of rapidly decreasing sequences, i.e.
$s=\{\xi=(\xi_j)_j\colon\,\,\sup_j|\xi_j|j^n<\infty\,\,\text{for all}\,\,n\in\mathbb{N}\}$ and $s'$ its topological dual, i.e.
...

**2**

votes

**0**answers

60 views

### The motivation of Weyl-Titchmarsh function

Given a second linear differential operator,
$(Hf)(x)=-\frac{d^2}{dx^2}(x)+V(x)f(x)$,
where $V$ is a bounded and
real valued function, $f$ lies in $L^2(\mathbb{R})$.
For an $z$ with $Im(z)\neq ...

**2**

votes

**0**answers

162 views

### Discrete p-Laplacian

One of the definitions of the discrete (weighted) $p$-Laplacian is the following:
$$\Delta_{p,w}u(x):=\sum_y |u(y)-u(x)|^{p-2}(u(y)-u(x))w(x,y).$$
Consider the one dimensional case. Then the free ...

**2**

votes

**0**answers

132 views

### Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as
$
\hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx.
$
The Hilbert transform ...

**2**

votes

**0**answers

83 views

### Semigroups on Banach Lattice

Let $\{Z(t)\}_{t\geq 0}$ be a semigroup of positive operators on a Banach lattice $X$. I want to show that
$$\int_0^1[Z(1-s)+Z(1+s)]fds-f\in X_+,\quad f\in X_+$$
Where $X_+$ denotes the positive ...

**2**

votes

**0**answers

85 views

### Necessity of coercivity assumption in Minty's theorem

Minty's Theorem states that a bounded, Continuous, monotone and coercive function on a Hilbert space is surjective.
A function $f:H\to H$ is called coercive if
$$ \lim_{\|x\|\to \infty} \frac{\langle ...

**2**

votes

**0**answers

288 views

### Predual of a von Neumann algebra in terms of trace class operators

For a von Neumann algebra $\mathcal{A} \subseteq \mathcal{B(H)}$ where $\mathcal{B(H)}$ is the space of all bounded linear operators on the Hilbert space $\mathcal{H}$, there is a Banach space $ ...

**2**

votes

**0**answers

87 views

### Constant in Maximal sobolev regularity

We know the following evolution equation
\begin{equation}
\left\{
\begin{array}{llc}
v_t=A v+f,\\
v(0)=0.
\end{array}
\right.
\end{equation}
$A$ generates a bounded analytic semigroup on a Banach ...

**2**

votes

**0**answers

53 views

### Is the Szego projection on a codim-$k$ CR manifold an integral operator?

The Szego projection on a CR manifold $M$ is defined to be the orthogonal projection from $L^2(M)$ to the closed subspace $H^2(M),$ where
$$H^2(M) = \{f \in L^2(M)\ |\ \bar{\partial}_{b}f = 0\ ...

**2**

votes

**0**answers

75 views

### series representation for *un*bounded perturbations of semigroup generators

Let $A$ generate an analytic $C_0$-semigroup on a Banach space $X$ and $B$ be a relatively compact perturbation, i.e., $B$ is compact as an operator from $D(A)$ (with the graph norm) to $X$. Then ...

**2**

votes

**0**answers

252 views

### Versions of the spectral theorem

Since any $C^*$-algebra can be represented as an algebra of bounded operators $\mathcal{B(H)}$ on a Hilbert space $\mathcal{H}$, the spectral theorem applies to all $C^*$-algebras:
($*$) ...