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31 votes
3 answers
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When is an integral transform trace class?

Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator $$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$ the operator $K$ is Hilbert ...
Marc Palm's user avatar
  • 11.2k
18 votes
1 answer
564 views

Is the space of Hankel operators complemented in B(H)?

Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^*S=I\neq SS^*$. Then a bounded operator $T:H\to H$ is Hankel if and only if it satisfies $TS=S^*T$. Let ...
Yemon Choi's user avatar
  • 25.8k
12 votes
3 answers
881 views

Bibliographic request concerning an article by Bernstein and Robinson

Concerning the article "Bernstein, Allen R.; Robinson, Abraham. Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in finding ...
Mikhail Katz's user avatar
  • 16.6k
11 votes
1 answer
1k views

Quasi-nilpotent trace class operators as limits of nilpotents

In as yet unwritten work with T. Figiel and A. Szankowski we make an observation in a Banach space context that for Hilbert spaces reduces to: If $T$ is a quasi-nilpotent (i.e., has only $0$ in its ...
Bill Johnson's user avatar
  • 31.5k
11 votes
0 answers
344 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 $L^1(\mathbb{R})$; ...
ABIM's user avatar
  • 5,405
10 votes
2 answers
1k views

What function is this? -Counterexample found: it is not Lipschitz-

THE FRAMEWORK Let $0<\lambda\le1$ and consider $$ \Psi:(\Bbb R[X]_0,||\cdot||_{\lambda})\longrightarrow(\mathcal C^{\lambda}[0,1],||\cdot||_{\lambda}) $$ defined as $$ \Psi(p):=\sup_{0\le u\le\...
Joe's user avatar
  • 779
10 votes
1 answer
598 views

What happens if we rotate the kernel of an integral operator?

Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, ...
limanac's user avatar
  • 452
10 votes
0 answers
710 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 ...
Terry Loring's user avatar
  • 1,749
9 votes
2 answers
611 views

When does $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_{\mathrm{op}}$ hold?

Define the Frobenius norm of a matrix as $\left\Vert A \right\Vert_{\mathrm{F}}=\sqrt{\sum_{i,j} A_{ij}^2}$ and the operator norm as $\left\Vert A \right\Vert_{\mathrm{op}}=\sup_{x \not = 0} \frac{\...
Henry's user avatar
  • 93
9 votes
1 answer
202 views

Literature request: Schatten class difference of semigroups

Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily self-adjoint) such that $A, A+B$ are generators of strongly continuous one parameter semigroups $e^{-tA},e^{-t(A+B)}$...
folouer of kaklas's user avatar
9 votes
1 answer
322 views

Cauchy path integral as a linear operator: kernel and image?

Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path-...
Loïc Teyssier's user avatar
9 votes
1 answer
569 views

strong measurability question

Let $X$ be a separable Banach space and $\mathcal L$ the collection of bounded linear operators on $X$. The strong operator topology has the sub-basis $\{B_{x,y,\epsilon}\colon x,y\in X,\epsilon>0\}...
Anthony Quas's user avatar
  • 23.2k
9 votes
2 answers
778 views

Rellich's theorem from compact resolvent

On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into $...
anonymous's user avatar
9 votes
1 answer
669 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" ...
Rye's user avatar
  • 191
8 votes
2 answers
640 views

Does a random sequence of vectors span a Hilbert space?

Let $\mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $\mathcal{H}$ such that $P(v \perp h) < 1$ for all $h \in \mathcal{H}.$ Suppose we sample an infinite ...
J. E. Pascoe's user avatar
  • 1,429
8 votes
0 answers
299 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). $...
Keith McClary's user avatar
7 votes
3 answers
2k views

Essential spectrum of multiplication operator

Let $a\in \mathcal{L}(L^2([0, 1], \mathbb{R}))$ be a multiplication operator. I wonder whether there is any work on calculating its essential spectrum. Is there any way to explicitly compute its ...
potionowner's user avatar
7 votes
1 answer
415 views

Is there a “Closure-of-Range Theorem” for Banach spaces?

The classic Closed Range theorem states that for a linear bounded operator $T:X\to Y$ between Banach spaces, and its transpose $T^*:Y^*\to X^*$, the four conditions: $T(X)$ is $s$-closed; $T(X)$ is $...
Pietro Majer's user avatar
  • 60.5k
7 votes
1 answer
187 views

numerical radius of hyponormal operator

Let $F$ be a complex Hilbert space. We recall that an operator $S\in\mathcal{B}(F)$ is said to be hyponormal if $S^*S\geq SS^*$ (i.e. $\langle (S^*S-SS^*)z,z \rangle\geq 0$ for all $z\in F$). ...
Schüler's user avatar
  • 724
7 votes
1 answer
592 views

Fractional powers of an operator

What is the large class of operators for which one can define fractional powers? For example, we can consider an operator $A: D(A) \subset X \rightarrow X$, generator of an analytic semigroup on a ...
Migalobe's user avatar
  • 395
7 votes
1 answer
306 views

An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that $$ f^2=f $$ (that ...
limanac's user avatar
  • 452
7 votes
1 answer
403 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 ...
Quoka's user avatar
  • 185
6 votes
3 answers
452 views

Norm estimation of identity plus two non-commuting self-adjoint operators

This is a problem that I have been stuck for a few months. Let $X$ be a Hilbert space and $A:B:X\to X$ be two non-commuting semi-positive self-adjoint bounded linear operators. Is it true that $$\|(I+...
John's user avatar
  • 503
6 votes
3 answers
852 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 ...
santker heboln's user avatar
6 votes
1 answer
322 views

Self-adjoint operator

Assume that $B$ is a self-adjoint operator and $\alpha\in(0,1)$. I need a reference for the following equality $$B^{-\alpha}=\frac{\sin\alpha \pi}{\pi}\int_0^\infty \lambda^{-\alpha}(\lambda+B)^{-1}\...
user47005's user avatar
6 votes
0 answers
170 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 ...
Suslina  Tatyana's user avatar
5 votes
1 answer
1k views

Reference request: The resolvent is analytic in the resolvent set

I am busy reading through Taylor's paper Spectral Theory of Closed Distributive Operators. On page 192, he defines the resolvent and spectrum of $T$: Later on in the paragraph, he then proceeds by ...
user860374's user avatar
5 votes
2 answers
276 views

Dilation of bounded linear operators

Let $H$ be a Hilbert space, and let $A$ be a contraction (bounded linear operator of norm $\leq 1$) on $H$. I heard in a recent talk that there is a (apparently famous) result due to Sz-Nagy which ...
SKNEE's user avatar
  • 51
5 votes
1 answer
408 views

English translation of von Neumann's Algebra der Funktionaloperationen (1930)

Does anyone know if there exists an English translation of von Neumann's early work in operator theory, in particular the paper Zur Algebra der Funktionaloperationen und Theorie der normalen ...
Malcolm Langfield's user avatar
5 votes
1 answer
224 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 ...
F. Carbon's user avatar
  • 105
5 votes
4 answers
839 views

Norm bounds on spectral variation and eigenvalue variation

Let $A$ and $B$ be two matrices of eigenvalues $\lambda_i$ and $\mu_i$, respectively. The spectral variation of $B$ w.r.t. $A$ and the eigenvalue variation of $B$ and $A$ are, respectively, \begin{...
T. Amdeberhan's user avatar
5 votes
1 answer
301 views

Unbounded Component of the Fredholm Domain

Let $X$ be a Banach space and $T \in \mathcal L(X)$. The authors Engel and Nagel introduce in their book "One-Parameter Semigroups for Linear Evolution Equations" on p. 248 the concept of the ...
Adriano's user avatar
  • 301
5 votes
1 answer
1k views

Domain of square root of a self-adjoint positive operator [closed]

Let $A \geq 0$ be a densely defined self-adjoint positive operator on a Hilbert space $H$ obtained by Friedrichs extension, and let $Q$ be the densely defined quadratic form associated to $A$, that is,...
anonymous's user avatar
5 votes
1 answer
158 views

a matrix of Onsager-Kaufman vs Schwarz-Wu

In my earlier MO question, I was seeking for a proof for $\det A_{\infty}:=\det(I_{\infty}-M_{\infty}^2) =\sqrt[4]{1-x^2}$ where $M_n$ is the $n\times n$ matrix: $$M_n =\left[\frac{2i+1}{2(i+j+1)}\...
T. Amdeberhan's user avatar
5 votes
0 answers
163 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 ...
Benjamin's user avatar
  • 245
5 votes
0 answers
109 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 ...
Jochen Glueck's user avatar
5 votes
0 answers
179 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 $S$ ...
SMS's user avatar
  • 1,407
4 votes
5 answers
991 views

About the trace class operators and their motivation

What is the motivation for trace class operators? Can any body suggest the most general and standard reference that includes Schatten p class operators as well. I have following references ...
user43716's user avatar
4 votes
1 answer
279 views

Reference request: Baire's theorem for operator ranges

Let $F$ be a Banach space. A vector subspace $U \subseteq F$ is called an operator range if there exists a Banach space $E$ and a bounded linear mapping $T: E \to F$ such that $TE=U$. By a quotient ...
Jochen Glueck's user avatar
4 votes
1 answer
1k views

Does the Legendre-Fenchel transform/convex conjugate of strongly convex functions have any desirable properties?

It is well known in convex analysis that when a closed, proper, function $f$ is Legendre-type, that is, essentially strictly convex and essentially smooth, the Legendre transform yields a dual ...
Concu Bine's user avatar
4 votes
1 answer
532 views

Diffusion semigroup generated by Laplacian

Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P_t, t > 0$ denote the diffusion ...
anonymos's user avatar
4 votes
1 answer
498 views

Generator of a $C_0$-semigroup restricted to a subspace

Suppose we have a decreasing filtration of Banach spaces $(E_t)_{t\geq0}$, inclusions $V_{s+t,t}:E_{s+t}\to E_t$ and projections $P_{t,s+t}:E_t\to E_{s+t}$ such that $P_{t,s+t}V_{s+t,t}=I_{E_{s+t}}$. ...
Ollie's user avatar
  • 1,411
4 votes
1 answer
1k views

definition of accretive operator

A relation T with domain and range in a Hilbert space is said to be accretive if the transformation $ (T − \lambda)/(T + \bar \lambda\ ) $ with domain and range in the Hilbert space is contractive for ...
Koushik's user avatar
  • 2,106
4 votes
1 answer
92 views

Continuous section inside a family of rank-varying operators

Good morning everybody, my question is as follows: let $K$ be a compact set and assume $F:K\to L(\mathcal H,\mathbb R^{m+1})$ be a continuous map from the compact set $K$ to the space of linear ...
guido giuliani's user avatar
4 votes
0 answers
72 views

Need reference of books/papers which deals with Ternary Banach Algebras

I'm interested in learning about ternary Banach Algebras ( mainly ideal theory and tensor product) Can someone please recommend me some papers/ books/ notes which deals with mentioned topics? Thank ...
Math Lover's user avatar
  • 1,115
4 votes
0 answers
120 views

Reductive Operator Problem

In the 1972 paper ''An equivalent Formulation of the Invariant Subspace Conjecture'' Dyer, Pedersen, and Porcelli announce the following result: The Invariant Subspace Problem has a positive ...
Adi Tcaciuc's user avatar
4 votes
0 answers
361 views

Spectral mapping theorem

Rudin's book contains in chapter 10 a spectral mapping theorem for (self-adjoint) unbounded operators that respects the point-spectrum, in the sense that he shows $f(\sigma_p(T))=\sigma_p(f(T))$ for ...
Kinzlin's user avatar
  • 305
4 votes
0 answers
121 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, u\...
anonymous's user avatar
3 votes
2 answers
1k views

Lecture notes on semigroup theory for linear evolution equations

I am reading (or trying to read :)) "One parameter semigroups for Linear Evolution equations" by Klaus-Jochen Engel and Rainer Nagel. I was wondering if someone was aware of a good set of lecture ...
Marcos Rojo's user avatar
3 votes
1 answer
573 views

Reference request: Irreducible operators

I had asked this question on MSE but did not get any response. I would like some reference to books that talk about irreducible operators on Banach lattices and its properties. A quick google search ...
Mark's user avatar
  • 343