All Questions
Tagged with reference-request operator-theory
123 questions
31
votes
3
answers
5k
views
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 ...
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 ...
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 ...
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 ...
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 $L^1(\mathbb{R})$; ...
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\...
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)$, ...
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
...
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{\...
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)}$...
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-...
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\}...
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 $...
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" ...
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 ...
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). $...
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 ...
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 $...
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$).
...
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 ...
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
$$
(that ...
7
votes
1
answer
403
views
Why are we interested in operators that share a basis of eigenfunctions?
I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context.
I ...
6
votes
3
answers
452
views
Norm estimation of identity plus two non-commuting self-adjoint operators
This is a problem that I have been stuck for a few months.
Let $X$ be a Hilbert space and $A:B:X\to X$ be two non-commuting semi-positive self-adjoint bounded linear operators. Is it true that
$$\|(I+...
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 ...
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}\...
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 ...
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 ...
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 ...
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 ...
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 ...
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{...
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 ...
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,...
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)}\...
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 ...
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 ...
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 $S$ ...
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
...
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 ...
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 ...
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 ...
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 $P_{t,s+t}V_{s+t,t}=I_{E_{s+t}}$. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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\...
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 ...
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 ...