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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2
votes
1answer
132 views

The operator equation $AB = \lambda BA$ for self-adjoint operators

Suppose that $A$ and $B$ are self-adjoint bounded linear operators on a Hilbert space and $\lambda \in \mathbb{C}$. It turns out that if $\lambda \notin \{-1, 1\}$ then $AB=\lambda BA \implies AB = ...
1
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0answers
36 views

Is $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$ completely contractive?

Take Hilbert spaces $H$ and $K$. Consider a linear map $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$. Is it true that $F$ is completely contractive? If it is, I would be very grateful ...
2
votes
0answers
38 views

Holomorphic semigroups vs analytic semigroups

Is there any difference between the two notions in the theory of semigroups? In the literature, we find some monographs use the farmer while others use the latter. I expect that they are always the ...
6
votes
3answers
628 views

Are nuclear operators closed under extensions?

Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram $\require{AMScd}$ \begin{CD} 0 @>>> X_1 @>f_1>> X_2 ...
3
votes
1answer
72 views

Generalized tensor-train decomposition

If $U\in\bigotimes_{k=1}^p\mathbb R^{n_k}$ and $U^{(k)}\in\mathbb R^{r_{k-1}}\otimes\mathbb R^{n_k}\otimes\mathbb R^{r_k}$ with$^1$ $$U_{i_1,\:\ldots\:,i_p}=\sum_{j_0=1}^{r_0}\cdots\sum_{j_p=1}^{r_p}U^...
1
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0answers
56 views

An elementary proof of Davies' inequality

In the paper Lipschitz continuity of functions of operators in the Schatten classes, Davies proved the following matrix inequality. Let $a_i,b_i>0$ for $1\leq i\leq n$ and $A$ be an $n\times n$ ...
0
votes
0answers
37 views

Tensor contraction (vector-valued trace) on $\bigotimes_{i=1}^k\mathcal L(E_{i-1},E_i)$

If $E_i$ is a $\mathbb R$-vector space, then the vector-valued trace $\operatorname{tr}_{E_1}:(E_2\otimes E_1^\ast)\otimes(E_1\otimes E_0)\to E_1\otimes E_0$ (or tensor contraction) is the ...
0
votes
0answers
45 views

Lower bounds in the space of compact operators

Let $H$ be a separable Hilbert space, and $K(H)$ the corresponding space of compact operators. Consider the "unit sphere" $S:=\{T\in K(H)|T\geq 0\text{ and }||T||=1\}$. Is it true that, given any pair ...
9
votes
2answers
248 views

Exponential decay of voltage potential difference

Consider the following adjacency matrix of a complete graph: $$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$ with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
1
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0answers
36 views

Operational quantities characterizing upper semi-Fredholm operators

An operator $T:X\rightarrow Y$ is said to be upper semi-Fredholm if its range is closed and its kernel is finite-dimensional. M. Schechter (1972) introduced a quantity $$\nu(T):=\sup_{\operatorname{...
0
votes
1answer
30 views

Difference of hypercyclic operator and identity

Let $B$ be a separable Banach space, $L:B \to B$ be a hypercyclic operator, $k>0$, $I_B$ the identity on $B$, and define $L_k: =k (I_B - L)$. When is $L_k$ hypercyclic on $B$? Can anything else ...
-1
votes
0answers
40 views

Relationship between negative operator eigenvalues

Let $L>0$, $c \in (-1,1)$ and $\varphi \in H_{per}^{2}([0,L])$ be fixed. Define $w:= 1-c^2>0$. Consider the matrix operator $\mathcal{L}_{R}: H_{per}^{2}([0,L]) \times L_{per}^{2}([0,L]) \...
1
vote
1answer
41 views

Uniform boundedness of resolvents on the imaginary axis

Let $A \colon \mathcal{D}(A) \subset \mathbb{H} \to \mathbb{H}$ be a closed linear operator in a Hilbert space $\mathbb{H}$, which generates a $C_{0}$-semigroup. Suppose that in a $\varepsilon$-...
2
votes
1answer
85 views

Is the reduced group $C^*$-algebra quasidiagonal

Let $G$ be an amenable group. I wonder whether it is true that the reduced group $C^*$-algebra $C_r^*(G)$ is quasidiagonal.
4
votes
1answer
90 views

Existence of an injective unbounded below operator

Given an infinite dimensional Banach space, can we always find an infinite dimensional Banach space $Y$ and an injective bounded operator $T:X\to Y$ such that $T$ is not bounded below? If $X^{*}$ is ...
1
vote
1answer
46 views

Determine the singular values of a compact operator in terms of the eigenvalues of an alternating tensor product of operators

Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be compact and $$|A|:=\sqrt{A^\ast A}$$ denote the square-root of $A$. By definition, the $k$th largest singular value $\sigma_k(A)$ of $A$ ...
6
votes
0answers
126 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 ...
7
votes
1answer
268 views

Why are we interested in operators that share a basis of eigenfunctions?

I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context. I ...
2
votes
1answer
112 views

Sum of strongly commuting self-adjoint operators

Let $A,B$ be two positive unbounded, self-adjoint operators on some Hilbert space that strongly commute. Let $D(A)$ and $D(B)$ denote their respective domain. Then, using for instance the spectral ...
3
votes
0answers
59 views

Bound on number of linearly independent eigenvectors of adjoint of composition operator

Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...
3
votes
0answers
49 views

Dense stratification of a separable Hilbert space

Let $\{X_i\}_{i \in \mathbb{N}} $ be a sequence of $n$-dimensional linear subspaces of the separable Hilbert space $H$ and let $\{\phi_i\}_{i \in I}$ be a sequence of continuous injective linear maps ...
5
votes
1answer
138 views

Approximating a projection by a sum of elementary tensors with a certain property

Let $A$ and $B$ be two C$^{*}$-algebras and suppose we have a non-zero projection $p\in A\otimes B$. (We can assume $A$ is nuclear, so that there is only one possible tensor product.) Does there ...
4
votes
1answer
57 views

Two measures of noncompactness of operators

Let $X,Y$ be Banach spaces. We denote by $\mathcal{K}(X,Y)$ the space of all compact operators from $X$ into $Y$. For an operator $T:X\rightarrow Y$, we let $$\|T\|_{e}:=\inf\{\|T-K\|:K\in \mathcal{K}(...
4
votes
1answer
123 views

Elliptic estimates for self-adjoint operators

Let $A$ be a symmetric matrix in $\mathbb R^n$ such that $A$ is positive definite and hence satisfies $0< \lambda \le A \le \Lambda < \infty.$ Let $T$ be a densely defined and closed operator ...
2
votes
0answers
69 views

A quantitative characterization of bounded approximation property

Recall that a Banach space $X$ has the approximation property (AP for short ) if for every compact subset $K$ of $X$ and every $\varepsilon > 0$, there exists a finite rank operator $S$ on $X$ such ...
3
votes
1answer
93 views

When is rank-1 perturbation to a positive operator still positive?

Let $A : \mathcal{H} \to \mathcal{H}$ and $B : \mathcal{H} \to \mathcal{H}$ be trace-class (hence compact) Hermitian operators on a separable Hilbert space. Assume that $A$ is strictly positive and ...
10
votes
1answer
284 views

Is the spectrum of a “self adjoint” operator real on $\ell^p$?

There might be an obvious answer to the question, but it doesn't come to mind. Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all ...
1
vote
1answer
127 views

Coercivity of linear operators

I am trying to understand the following argument: Let $\mathcal{L}:L^2(\mathbb{R})\to L^2(\mathbb{R})$ be an essentially self-adjoint unbounded linear operator with domain $D(\mathcal{L})=H^s(\mathbb{...
0
votes
1answer
120 views

Injectivity of an integral operator

Consider the operator $$K:L^2(0,1)\rightarrow L^2(0,1) \\ u\rightarrow\int_0^1k(s,x)u(s)ds.$$ with $k\in L^2((0,1)\times(0,1)).$ I want to know under what assumption the kernel is reduced to zero. i....
5
votes
0answers
44 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 ...
7
votes
1answer
315 views

Spectrum of “classical” operators

Lately, I've been reading a couple of papers from different one-dimensional PDE contexts on which operators like $\mathcal{L}:=-\partial_x^2+c_*+\Phi$ repeatedly appear. Usually, on these contexts $\...
1
vote
0answers
49 views

On various versions of the harmonic oscillator

The standard $n$-dimensional harmonic oscillator is the operator $ \mathcal H=\frac{1}{2}\sum_{1\le j\le n}(D_j^2+x_j^2), \text{ $D_j=-i\partial_{x_j}$}, $ and its spectral decomposition is $$ \...
2
votes
0answers
67 views

First Dirichlet eigenvalue below second Neumann eigenvalue?

Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary. I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...
1
vote
0answers
38 views

Subgradient chain rule

Suppose $$F:\mathbb{R}^n \to \mathbb{R},\; F(x)=\mathrm{max}_\mathrm{eig}(C-\mbox{diag}(x)).$$ I am trying to find a subgradient of $F$ at $x_0$. A subgradient of $\mathrm{max}_\mathrm{eig}$ is given ...
3
votes
0answers
34 views

Continuity of local spectral radius

Let $H$ be a complex Hilbert space and let $T \in \mathcal{B}(H)$ be a linear, bounded operator. Given $x \in H$ we define its local spectral radius as $$r_T(x) = \limsup\limits_{n\rightarrow\infty} \|...
3
votes
0answers
64 views

Integral kernel of composition of linear operators acting in $L^2(\mathbb{R})$

Let $A$ and $B$ be linear operators acting on $L^2(\mathbb{R})$ or some dense subset of that space. We assume that they are integral operators (possibly with distributional kernel) $Af(x)=\int_{\...
1
vote
1answer
109 views

Spectral properties of operators mapped to zero by some polynomial

Let $T$ be a bounded operator on a Banach space $X$ and suppose that there is a non-constant polynomial $p$ such that $p(T) = 0$. It seems to be well known that the spectrum of such an operator ...
8
votes
0answers
176 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 ...
1
vote
1answer
73 views

Explanation of $\sigma$-weak topology von a von Neumann algebra [closed]

Let $A$ be a von Neumann algebra. I want to understand the precise meaning of the $\sigma$-weak topology on $A$. What I understand so far is the following: The $\sigma$-weak topology, which we will ...
2
votes
2answers
146 views

Are there any techniques that can be used in the case when a Neumann series doesn't converge?

Suppose we have a bounded linear operator $A = A(\gamma):H_1\to H_2$ where $H_1$ and $H_2$ are Hilbert spaces and $\gamma>0$ is some parameter, and we are interested in the solution to $$ (I-A)x = ...
0
votes
0answers
47 views

Support of commuting elements in a von Neumann algebra

Let $\mathcal M$ be a semifinite von Neumann algebra and $a$ be an unbounded positive self-adjoint operator affiliated to $\mathcal M$. Suppose $x\in\mathcal M$ is such that $x$ commutes with all ...
0
votes
0answers
37 views

Limit of spectral projection of increasing sequence of positive operators

Let $\mathcal M$ be a von Neumann algebra. Suppose $(x_\alpha)\subset \mathcal M$ is bounded increasing net of positive operators converging to a positive operator $x\in\mathcal M.$ Is it true that $\...
6
votes
0answers
79 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\...
1
vote
1answer
153 views

Sequence of Hilbert Schmidt operators

Consider the Banach space $\mathcal K=S_2(H)$ of Hilbert Schmidt operators on a Hilbert space $H$. I am looking for an example of two pairs of sequences $\{T^{(i)}\},\{\tilde T^{(j)}\}$ and $\{S^{(i)}\...
4
votes
0answers
181 views

Discrete superharmonicity

The value at $(n,m)$ of the “Discrete Laplace operator” (see wikipedia) of a function $f$ in $\Bbb Z\times \Bbb Z$ is $\Delta f(n,m)= \frac{1}{4}( f(n+1,m)+f(n,m+1)+f(n-1,m)+f(n,m-1))-f(n,m)$: the ...
2
votes
1answer
81 views

If $\|S\|<\sin\frac{\pi}{2n}$ then $\|P(I-S)^ku\|\neq 0$ for all $k=0,\ldots,n$

I want to show the following: Let $H$ be a Hilbert space and let $S:H\to H$ be a bounded operator such that $$\|S\|<\sin\frac{\pi}{2n}.$$ Let $\mathcal{L}$ be a closed subspace of $H$ and $$u_k:=...
0
votes
0answers
81 views

Numerical range of tensor product of two matrix

Let $T\in M_n$. Is the following true $$\bigcap\limits_{B\in M_2\\\text{tr}(B)=0}\left\{X\in M_2: W(X)\subseteq W(T)\text{ and } W(B\otimes X)\subseteq W(B\otimes T)\right\}\subseteq\bigcap\limits_{...
3
votes
1answer
133 views

reference request: unbounded operators on normed spaces

I'm looking for books where the theory (basic properties, adjoints etc.) of unbounded linear operators between locally convex spaces or at least Banach spaces is developed. In Brezis' functional ...
1
vote
0answers
78 views

A question on the Dieudonné property

Recall that a Banach space $X$ is said to have the Dieudonné property if for every Banach space $Y$, an operator $T:X\rightarrow Y$ that transforms weakly Cauchy sequences into weakly convergent ...
3
votes
0answers
143 views

Completely positive, unital maps acting on unitary operators [solved]

Call a completely positive, irreducible, unital map $E$ on the operators on a finite-dimensional Hilbert-space primitive if there is only a single eigenvalue with modulus 1 (all others have modulus &...

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