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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Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?

In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$. The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), but it may fail to have a bounded ...
Pietro Majer's user avatar
2 votes
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Convergence of eigenfunctions

In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
builtdifferential's user avatar
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Example of an $H^1$ function on the bidisk that is not a product of two $H^2$ functions

Fix $n \in \mathbb{N}$ and consider the Hardy space $H^1 := H^1(\mathbb{D}^n)$, consisting of holomorphic functions $f$ on the unit polydisk $\mathbb{D}^n=\mathbb{D}\times\dots\times\mathbb{D}$ such ...
EG2023's user avatar
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Koopman operators on $L^p(X)$

On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ ...
Scottish Questions's user avatar
1 vote
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Spectral gap of a Markov operator on $L^2$ with a symmetric $L^\infty$ kernel

Let $I$ be a compact interval, say $I:=(0,1)$, and $k\in L^\infty(I\times I)$ a symmetric Markov kernel, i.e. $k(x,y)=k(y,x)$ and $$\int_I k(x,y) d y = 1\qquad\mbox{for almost all } x\in I.$$ Let $K:L^...
Johny B's user avatar
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Infinite tensor product of Hilbert spaces [duplicate]

Recently while reading an article I came across the usage of infinite tensor product of Hilbert spaces. I have got a basic understanding of doing computations in infinite tensor product while reading ...
Jake's user avatar
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A question about associated operator on continuous functions space equiped with L2 norm

$$\text{For M a connected compact manifold, T is in }C^{1+\nu}(M,M) \text{ with } \nu\in(0,1),\\ \text{i.e. DT is some Hölder continuous function with Hölder exponent }\\ \text{, Denote m as the ...
WaoaoaoTTTT's user avatar
2 votes
0 answers
159 views

What are alternative or equivalent definitions of a positive-definite function on a group?

The standard definition of a positive-definite function on a group goes as follows: Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...
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Unitary versus isometric operators

Let $\mathbb H$ be a Hilbert space, and let $\mathcal B(\mathbb H)$ be the space of bounded operators on $\mathbb H$, equipped with the operator-norm topology. Let $\mathbb R\ni t\mapsto A(t)\in \...
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Looking for examples of kernels with scalar Pick property but not the complete Pick property

I am studying Pick Interpolation and Hilbert Function Spaces by Agler and McCarthy. A kernel $k$ on a set $X$ is said to have $M_{s,t}$ Pick property whenever $x_1,x_2, \ldots , x_n \in X$ and $W_1, ...
ashK's user avatar
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1 answer
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Matrix-order derivatives (differentiating a function a matrix number of times)

I have been exploring methods of generalizing the order of derivatives to a broader range of inputs (such as real numbers, complex, and now matrices). We are very well familiar with integer-order ...
charlesalexanderlee's user avatar
1 vote
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22 views

Characterization of the Picard's condition for integral equation

Picard's condition (Thm. 15.18, Kress et al. 1989) is essential to study the existence of a solution of a Fredholm integral equation of the first kind. Specifically, consider (the univariate case) the ...
Mingzhou Liu's user avatar
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What is the current status of research on the von Neumann's inequality for $n \ge 3$?

Problem Let $(T_1, \ldots, T_n)$ be a tuple of commuting contractions in Hilbert space $H$. Does a constant $C_n \ge 1$ exist, for which it would be true, that: $$\forall_{p \in \mathbb{C}[x_1, \ldots,...
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If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?

Definitions Representation Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$. We call $\...
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What is meant by saying that the Shilov boundary of the polydisc $\mathbb D^n$ is $\mathbb T^n\ $?

Let $A$ be a complex Banach algebra and $M_A$ be the space of all non-zero multiplicative linear functionals on $A$ equipped with the weak$^*$-topology. Let $\widehat A$ be the image of $A$ under the ...
Anacardium's user avatar
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Why is von Neumann inequality important for equivalence of $\forall_j \ T_j^n\rightarrow 0$ in A-topology and abs continuity of $(T_1,\ldots, T_N)$?

The whole theorem goes as follows: Let $(T_1, \ldots, T_N)$ be a tuple of commuting operators in Hilbert space $H$ satisfying: $$\exists_{M > 0} \ : \ \forall_{p \in \mathbb{C}[z_1, \ldots, z_N]} \ ...
S-F's user avatar
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Trace class operators

There is a notion of trace class operator in a Hilbert space. Is there a notion of trace class operator in arbitrary Banach space? locally convex space? A reference will be helpful.
asv's user avatar
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1 vote
1 answer
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A certainty principle?

Let $\mathcal{A}$ be a unital $\mathrm{C}^*$-algebra with $\varphi\in\mathcal{S}(\mathcal{A})$ a state. Where $$\sigma_\varphi(a):=\sqrt{\varphi((a-\varphi(a)1_{\mathcal{A}})^2)}\qquad (a\in \mathcal{...
JP McCarthy's user avatar
2 votes
1 answer
225 views

Are these conditions regarding products of consecutive terms in a sequence of positive numbers equivalent?

Assume $w_n$ is a bounded (weight) sequence of positive numbers. We want to consider products of consecutive terms in this sequence. For $i,j\in \mathbb{N}$, define $M_i^j = w_i w_{i+1}\cdots w_{i+j-1}...
David Walmsley's user avatar
2 votes
0 answers
137 views

How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?

Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, Is $\operatorname{Id}-K$ a proper map? I think maybe it has ...
boundary's user avatar
2 votes
0 answers
145 views

What are the current open problems in dilation theory?

I just started doing my PhD in mathematics. My topic is unitary dilations of operators. I've been reading a lot of papers on that subject so far (especially about the dilation of $n \ge 3$ commuting ...
S-F's user avatar
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Zeta zeros and prolate wave operators

Recently, Connes, Consani and Moscovici in https://arxiv.org/abs/2310.18423 have blended two of their results on zeta zeros and the prolate wave operators, which, they say, "suggests the ...
Jon23's user avatar
  • 97
1 vote
1 answer
141 views

Moore–Penrose inverse of the square root

I'm seeking the Moore–Penrose inverse of the square root (if it exists) for a generalized invertible operator T. I've looked into whether it's the square root of its Moore–Penrose inverse but haven't ...
Phd m's user avatar
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2 votes
0 answers
88 views

Self adjoint operators from energy functionals

It is known that the equation $$ \Delta f = 0 $$ on some bounded domain $\Omega$ on $\mathbb{R}^n$ subjected to certain boundary conditions can be derived through the minimization of the Dirichlet ...
user8469759's user avatar
2 votes
1 answer
164 views

On spectral calculus and commutation of operators

Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...
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Operator identity

Let $T:\mathcal{D}(A)\to\mathcal{H}$ be a unbounded, self-adjoint, operator with positive spectrum $\sigma(T)\subset [\varepsilon,\infty)$ for $\varepsilon>0$. Hence $T$ is bijective with bounded ...
B.Hueber's user avatar
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5 votes
2 answers
390 views

Logarithm of a bounded operator

Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a closed half-line issued from 0 in the complex plane. Then I guess that $ A=\exp L $...
Bazin's user avatar
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0 answers
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Characterizing isometry invariant functions

I am trying to understand functions invariant under isometries better. In particular functions with two inputs. I.e. $$ C(x,y) = C(\phi x, \phi y) $$ for any isometry $\phi$. The motivation are ...
Felix B.'s user avatar
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1 vote
1 answer
231 views

An example of non-invertible operator $F$ such that $P_nF$ is invertible on $\operatorname{Im}P_n$ or proving that It is impossible

Given: $X$ - any Banach space $F : X \to X$ (linear bounded and non-invertible) $P_n$, which is projector that strongly converges to the identity operator $I$ as $n \to\infty$ Can you help me come ...
TorteDeline's user avatar
0 votes
0 answers
159 views

Connection between number theory and operator theory

I was wondering if there is any connection between number theory and operator theory. Especially the applications of Hardy spaces, de branges-Rovnyak spaces, Dirichlet spaces in number theory. For ...
M.P's user avatar
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1 vote
1 answer
41 views

Wold decomposition of toral endomorphisms

Suppose that $A\in M_d(\mathbb{Z})$ is a $d \times d$ matrix with non zero determinant and suppose that $\mathbb{T}^d$ is the $d$-dimensional torus. Then one can define an operator on $L^2(\mathbb{T}^...
an_ordinary_mathematician's user avatar
1 vote
1 answer
102 views

weakly separated sequences in RKHS are separated by Gleason metric

I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, the authors ask to observe that weakly separated in a Reproducing kernel hilbert space implies separated ...
ashK's user avatar
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1 vote
0 answers
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Interpolating sequences are strongly separated

I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, titled "Interpolating Sequences", the authors say that every interpolating sequence is ...
ashK's user avatar
  • 83
3 votes
2 answers
195 views

Domain of spectral fractional Laplacian

Let $(M,g)$ be a complete Riemannian manifold with Laplacian $\Delta:C^{\infty}_{c}(M)\to C^{\infty}_{c}(M)$ (think of $\mathbb{R}^{d}$ if you wish). This operator is essentially self-adjoint in $L^{2}...
B.Hueber's user avatar
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4 votes
0 answers
215 views

On the Dunford-Pettis property and multiplier algebras

I am not an expert in operator algebras, so if the answer to this question might be trivial, that might be one reason for that: Let $\mathcal{A}$ be a $C^\ast$-algebra. Then $\mathcal{A}^{\ast \ast}$ ...
Alexander Dobrick's user avatar
0 votes
1 answer
186 views

Perturbation of matrices

Let $A(t)$ be a symmetric $n\times n$ matrix that continuously depend on $t\in [0,1]$. Let $\lambda_1(t)$ stand for the smallest eigenvalue for $A(t)$. Question. Does there exist a Lebesgue measurable ...
Ali's user avatar
  • 3,995
1 vote
0 answers
108 views

Continuity of a minimizing measure w.r.t a parameter

Let $V_t(x)=x^2+t\phi(x)$ where $t>0$ and $\phi\in C^\infty_c(\mathbb{R})$. My question is what can be said about the continuity of the (unique) minimizer (among probability measures) of the ...
BlueCharlie's user avatar
2 votes
1 answer
116 views

$K_0$ group of an infinite factor

The following question was already posted in this link but I could not understand hints given in this post. Let $\mathcal{M}$ be an infinite factor and my question is how to prove that $K_0(\mathcal{M}...
Sanae Kochiya's user avatar
2 votes
0 answers
113 views

Self-adjointness of fractional laplacian

Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
B.Hueber's user avatar
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2 votes
0 answers
60 views

Proof of the convergence of the Rayleigh-Ritz Method?

In this article The convergence of the Rayleigh-Ritz Method in quantum chemistry by Bruno Klahn & Werner A. Bingel they have at page 11 Let $H_B$ be that Hilbert space which can be obtained as the ...
amilton moreira's user avatar
1 vote
0 answers
76 views

Is there a way to linearize matrix quadratic forms?

Say $x$ is a random vector in $\mathbb{R}^n$. Then, given a (deterministic) symmetric real positive definite matrix $A$, if we want to calculate the expectation of the quadratic form, we can use the ...
Drew Brady's user avatar
6 votes
0 answers
227 views

Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?

QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, satisfying $\sup_{x\in G} \lVert x\rVert <\infty$, whose elements are all of the form "identity+compact" (sometimes called &...
Yemon Choi's user avatar
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2 votes
0 answers
158 views

Approximating $L^p$ functions by eigenfunctions of Laplacian

I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932. In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
ze min jiang's user avatar
0 votes
0 answers
126 views

Question about a step in the proof of the min-max principle

I honestly do not think this is a hard question, maybe it is even obvious but I tried MSE and had no success so far, so I am reproducing the question Question about the proof of the min-max principle ...
MathMath's user avatar
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0 votes
0 answers
38 views

Is there any generalization of the convexity of $t^{-p}$ for $p > 0$ for real symmetric positive definite matrices?

Let $p > 0$. On the positive reals, $t \mapsto t^{-p}$, is a convex function, as can be seen easily by a plot or differentiation. However, unfortunately, unless $p \in (0, 1]$, the map $f_p(X) = X^{...
Drew Brady's user avatar
-2 votes
1 answer
133 views

Relationship between noncommutative torus for different values of theta [closed]

Let $u,v\in B(L_2(\mathbb T))$ defined as $u(f)(z)=zf(z)$ and $v(f)(z)=f(ze^{-2\pi i\theta})$ for $z\in\mathbb T$ where $\theta\in\mathbb R\setminus\mathbb{Q}$. Denote the $C^*$ algebra generated by $...
A beginner mathmatician's user avatar
5 votes
2 answers
419 views

Reconstruction of second-order elliptic operator from spectrum

Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous ...
Math_Newbie's user avatar
6 votes
1 answer
192 views

Existence of adjoint operators on manifolds

Let $(M,g)$ be an oriented Riemannian manifold and $V$ a finite-rank vector bundle equipped with a non-degenerate bundle metric $\langle\cdot,\cdot\rangle_{V}$. This bundle metric, in turn, gives rise ...
G. Blaickner's user avatar
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0 votes
0 answers
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A question regarding certain sequences in hyperfinite type $II_1$ factor

Let us consider the hyperfine type $II_1$ factor $\mathcal M$ arising from the inclusion $M_2\subseteq M_{2^2}\subseteq \dots M_{2^k}\subseteq\dots$ of matrix algebras with the normalised trace $\tau$....
A beginner mathmatician's user avatar
2 votes
0 answers
84 views

How to show $ |(Bx,x)|\leq (Ax,x) $ for any $ x\in D(A) $ here?

On the Hilbert space $ H $, $ A $ is a non-negative self-adjoint operator and $ B $ is a symmetric operator. Let $ D(B)\supset D(A) $, where $ D(A) $ and $ D(B) $ are definite domain for $ A $ and $ B ...
Luis Yanka Annalisc's user avatar

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