Questions tagged [operator-theory]
Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.
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Boundedness of a matrix operator in a norm
I would like to ask a simple question.
How do I show that a matrix is a bounded linear operator, for example this matrix
$$A=\begin{bmatrix}
t~~0\\
0~~\frac{1}{\sqrt{t}}
\end{bmatrix}$$
I know that ...
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$T$ compact operator, Let $\Delta^*_{\bar{\lambda}}$ subspace of $X^*$. Prove $\Delta^*_{\bar{\lambda}} = \bar{\Delta^*_{\bar{\lambda}}}$
We have proved the following claim:
Let T be compact operator and $\lambda \neq 0$. Then $\Delta_\lambda = \bar{\Delta _\lambda}$.
Now there is the corollary:
$T$ compact operator, Let $\Delta^*_{\...
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Question about invariant subspaces of C*-algebras.
Let $M$ be a subspace of some Hilbert space $H$ and let $U$ denote a $C*$-algebra contained in $L(H)$. Is it true that closure$[UM]$ is an invariant subspace of $U$?
I believe it is:
Assume $x \in$ ...
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Almost orthogonal operators after a relative scaling
If two positive operators $Q_1$ and $Q_2$ with unit $\ell_1$ norm are almost orthogonal: $\parallel Q_1 - Q_2 \parallel_1 \geq 2 -\epsilon$, then what can we say about the operators $Q_1$ and $c Q_2$, ...
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Multiplicative Functionals: Intuition
I am trying to get a better intuition on multiplicative functionals over a commutative Banach algebra. The definition to be is clear (simply: unital algebra homomorphisms into $\mathbb{C}$) and I am ...
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Definition of Hilbert-Schmidt integral operator
In the definition of Hilbert Schmidt integral operator, we require the kernel $K(\cdot,\cdot)$ to be defined on $L^2(X\times X)$. I don't quite understand this restriction. Can we allow $K$ defined on ...
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Multiplicity of bilateral shift on a Banach space
Let $\mathbb{X}$ be a Banach space. A bijective linear map $V: \mathbb{X} \to \mathbb{X}$ is said to be a bilateral shift if there is a closed subspace $\mathbb{L}$ of $\mathbb{X}$ such that
$\mathbb{...
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Whether we can get energy estimate $d\|X\|^2+2\|X\|_{H^1}^2 dt=2(X, B dW) +{\rm Tr}(BB^*)dt$ of a weak solution of stochastic differential equations? [closed]
Set an probability space $(\Omega,{\mathcal F},P)$ with filtration ${\mathcal F}_{t,t \ge 0}$, and
seperatable Hibert space $H$ and $U$
a self adjoint, sectoral，densely defined operator A on H with ...
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Commutators of unbounded operators on Hilbert spaces
Commutation seems to be a tricky business when it comes to unbounded operators, because of the domain questions. I have some trouble understanding the usual material about commutators of unbounded ...
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Proving an Inequality in Hilbert Space: $\int_0^1 \chi_{[t,\infty)}(T) dt \le T$ for $T\ge 0$
Let $\mathcal H$ be a Hilbert space and $\mathcal B(\mathcal H)$ denotes the set of all bounded operators on $\mathcal H$. An element $T\in \mathcal B(\mathcal H)$ is positive, we write $T\ge 0$ if $\...
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Basis of eigenfunction of nonself-adjoint operator
If $H$ is a separable Hilbert space, $A$ is a bounded nonself-adjoint operator, $\{\lambda_n\}_{n\in\mathbb{Z}}$ are the eigenvalues of $A$, and the corresponding eigenfunctions are $\{\psi_n\}_{n\in\...
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Why is the von Neumann inequality not always fulfilled for n-tuples of commuting contractions? Why cant we just take the single dilations and get it?
Let $(T_1, \ldots, T_n)$ be an $n$-tuple of contractions in a Hilbert space $H$. We know, that for every single $T_i$, there exists a unitary operator $U_i$ in a Hilbert space $K_i$, such that $$T_i = ...
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Is my formula for this projection correct?
Let $\phi \in L^{2}(\mathbb{R}^{d})$ be fixed. Denote by $P$ the orthogonal projection onto the subspace orthogonal to $\text{span}\{\phi\}$. In other words, for $f \in L^{2}(\mathbb{R}^{d})$ set:
$$(...
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Does there exist a widely-used operator $\boxdot$ such that $(\theta \boxdot \phi)(x) := \theta(x) \circ \phi(x)$?
Let $\forall X,Y : L(X,Y)$ symbolize the set of all linear operators from $X \rightarrow Y$.
Let us have operator-valued functions $\theta : I \rightarrow L(Y,Z)$ and $\phi : I \rightarrow L(X,Y)$.
It ...
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Stability of Subspaces under a Linear Map in Direct Sum Decomposition
Consider the vector spaces $D_1$, $D_2$, $D$ and $X$ such that $D\subset X$ and $D=D_1\oplus D_2$.
Furthermore, suppose that $L:X\longrightarrow D$ is a linear map such that $D_1$ is stable under $L$...
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Lower bound of integral operator in $L^{\infty}$
Let $\mu$ and $\nu$ be two $\sigma$-finite measures, and consider the operator (supposed well-defined) $L^{\infty}(\mu)$ to $L^{\infty}(\nu)$ by $Tg(y) = \int T(x,y)g(x) \mu(dx)$ where the kernel $T(x,...
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Index of Callias operator and application in physics
In his article "Axial Anomalies and Index Theorems on Open Spaces" (https://link.springer.com/article/10.1007/BF01202525) C.Callias shows how the index of the Callias-type operator on $R^{n}$...
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Are creation and annihilation operators special?
In Weinberg's The Quantum Theory of Fields,volume I, the author quotes a theorem that left me a bit mystified. He states
Any operator $O: \mathscr{H} \rightarrow \mathscr{H}$ may be written
$$O=\sum_{...
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Kraus operators
Suppose we have a POVM given by the family of positive, hermitian operators $\{E_i\}_{i\in I} \in \mathcal{H}$.
From the Neimark dilation theorem we know that the given POVM can be obtained from ...
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Is the resolvent of a local operator local?
Let $A$ denote a bounded linear operator on the Hilbert space $l^2(\mathbb{Z})$. We call $A$ a local operator if and only if there exists a $C \geq 0$ such that $\langle e_x | A | e_y \rangle = 0 $ if ...
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Unique extension of $*$-representation into an abstract multiplier algebra
I'm trying to find a proof of the following fact:
Let $A,B$ be $C^{*}$-algebras and $\pi: A \longrightarrow M(B)$ be a non-degenerate homomorphism in the sense that $\pi(A)B$ densely spans $B$. Then ...
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let $\phi : B(l^{p}(X)) \to B(l^{p}(Y)) $ be an isomorphism, does $\phi$ necessarily preserve rank of operators?
let $l^{p}(X)$ and $l^{p}(Y)$ be some $l^p$ function spaces, $B(l^{p}(X))$ and $B(l^{p}(Y)) $ be bounded linear operators on themselves, $\phi : B(l^{p}(X)) \to B(l^{p}(Y)) $ be an isomorphism as ...
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Limiting behavior of integral representation of $(\sqrt{\alpha^2-\partial_x^2}-\alpha)f(x)$
While studying pseudo-differential operators of type $\left(\sqrt{\alpha^{2} - \partial_{x}^{2}}-\alpha\right)\operatorname{f}\left(x\right)$, I came across the following integral representation of ...
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Bergman projection maps $L^q \left (\mathbb D^2 \right )$ boundedly onto $\mathbb A^q \left (\mathbb D^2 \right )$ for any $q \geq 2.$
Let $\mathbb A^2 \left (\mathbb D^2 \right )$ be the Bergman space consisting of square integrable holomorphic functions on $\mathbb D^2$ and $\mathbb P : L^2 \left (\mathbb D^2 \right ) \...
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Heat semigroup on $C_b(\mathbb R)$
Let $(X,\|\cdot\|)\in \{(L^2(\mathbb R),\|\cdot\|_{L^2}),(L^\infty(\mathbb R),\|\cdot\|_{L^\infty}),(C_b(\mathbb R),\|\cdot\|_{\infty})\}$
I have a question regarding the heat semigroup $$T_tf:=(\...
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About evolution problem with variable coefficients.
I'm studying differential operators. For example, in the evolution equation
\begin{align}
u_t&=(1-\partial_x^2)u\\
u(0)&=u_0
\end{align}
Question 1. Does this problem have any name in the ...
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Mean ergodic Operators [closed]
I am currently listening to some functional analysis lecture and solved the following exercise:
Let E be a Banach space, $T,S:E\rightarrow E$ bounded linear Operators, such that $\exists k \in \Bbb N:...
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Operator exponential equality question: Does $X(\sigma) = Y\sigma$ imply $\exp(X)(\sigma) = \exp(Y)\sigma$?
See this question and other linked questions I am exploring on physics stack exchange. https://physics.stackexchange.com/questions/819663/ad-circ-exp-exp-circ-ad-and-ei-theta-2-hatn-cdot-sigma-sigma-e-...
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Bishop's approximation theorem
I am trying to study the generalization that Axler made to Cuckovic work on the commutants of $T_{z^n}$ (Toeplitz operator with symbol $z^n$) on $L^{2}(\mathbb{D},dA)$, in the resource I am using, the ...
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Injectivity of Kernel Operator in Lp spaces
here is the context: Let $T$ be a kernel operator from $L^1(\mu)$ to $L^1(\nu)$ (probability measures in my problem), defined by
$ (Tf)(x) = \int f(y) \, k(x,y) \, \mu(dy). $
More generally, is there ...
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Basis of eigenfunctions and spectrum
I have a linear, closed and densely defined operator $A$ defined on a Hilbert space $H$. I prove that the point spectrum consists of discrete eigenvalues, and that the eigenfunctions of $A$ and of $A^*...
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Pure states in the $C^*$-algebra of compact operators
Let $H$ be a Hilbert space. I know the pure states of $B(H)$ whenever $H$ is finite-dimensional. They are the vector states $W_x$, i.e., of the form $W_x(A)=\langle Ax,x \rangle$, where $A\in B(H)$. ...
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Can an unbounded operator and its adjoint both have full domains?
Let $H$ be a complex Hilbert space. This post shows that there exist unbounded (which I will use to mean “not bounded”) operators on $H$ whose domain is all of $H$, i.e., $\mathcal D(T) = H$ (although ...
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Is the notion of compactness of operators preserved under quasi-similarity?
Let $H$ be a Hilbert space and $T \in \mathcal L (H).$ Then $T$ is said to be quasi-invertible if it is one-one and has dense range. The operator $T$ is said to be quasi-similar to an operator $S \in \...
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On self-adjoint extensions and multiplicity of eigenvalues
I hope you can help me with the following question.
Let $B$ be a densely defined closed symmetric operator on a infinite-dimensional separable complex Hilbert space $\mathcal{H}$ with deficiency ...
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Kernel dimension is preserved under uniform convergence of bounded operators on a Hilbert space
I hope you can help me with the following question. Consider a sequence $C_{n}$ of bounded operators on an infinite-dimensional separable complex Hilbert space $\mathcal{H}$ that converges uniformly ...
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Computing operator norm of convolution on bounded interval
Consider the space $X:=C([0,1],\mathbb R)$ of continuous real functions with inner product such that $\langle f,g\rangle:=\int_0^1f(x)g(x)\mathrm dx$.
And consider a continuous function $c$ from $[-1,...
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Operators that preserve RKHSness?
Are there any results on operators that preserve the reproducing property?
As an example, orthogonal projection preserves this property (and maps the reproducing kernel to the reproducing kernel as a ...
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What is wrong with this proof that a linear, bounded, time invariant operator on $L_p$ must be a convolution?
I'm trying to understand if this is true and how to prove it, "If $T$ is a bounded, time invariant operator on $L_p(\mathbb{R})$, then $T$ is a convolution operator.''
Here's an attempt at a ...
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What is a conjugate unitary operator?
I'm studying operator theory and I have encountered the concept of a cojugate unitary operagtor several times. However, I cannot find any reliable references. There is one paper which claims that a ...
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When does the normality of $T^3$ imply that of $T^2$, where $T\in B(H)$?
Let $T\in B(H)$. Assume that $T^3$ is normal, i.e., $T^{*3}T^3=T^3T^{*3}$. When is $T^2$ normal?
Here is a known related result, which may be more or less trivial:
If $T^2$ is normal and $T$ has a ...
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Prove that for every $n \in \mathbb{N}$, the mapping $A_n$ is a bounded linear operator from $C[0,1]$ to $C[0,1]$ and calculate its norm.
Let $C[0,1]$ be a normed space equipped with the norm $\|\cdot\|_\infty$, and let for every $n \in \mathbb{N}$, the mapping $A_n$ be given by the prescription $
(A_n(f))(x) =
\begin{cases}
f(x), &...
user1316790
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Uniqueness of solution of the operator equation $AXB=BXA$ on $B(H)$
Suppose we have Hermitian operators $A$ and $B$ in the infinite-dimensional Hilbert space $H$. I am interested in conditions under which the operator equation $AXB - BXA = 0$ has a unique Hermitian ...
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spectral theory of pseudo-differential operators of class $S^m$
I would be very grateful if you could give me the titles of books that deal with the spectral theory of pseudo-differential operators of class $S^m$
Thank you very much.
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On the compactness of the square of a finite double norm integration operator $T$ on $L^1 (\mu)$.
Good morning everyone,
I have read in S. P. Eveson, Compactness Criteria for Integral Operators in L∞ and L1 Spaces, Proceedings of the American Mathematical Society 123, 1995, 3709-3716 :
"If $(...
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Weighted shift operators with absolutely equal weights are unitarily equivalent
I have been stuck on the following exercise from Elementary Functional Analysis by Barbara MacCluer:
Exercise 2.10. Let $\{a_n\}_{n = 1}^\infty$ be a bounded sequence of complex numbers. Fix an ...
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Proof relies on the operator being a contraction...what is it in the statement of the lemma implies that the operator is a contraction?
I am trying to understand the following.
Lemma:
Let $H$ be a Hilbert space, let $\Phi = \{E_a\}$ be a linear map $B(H) \to B(H)$ where $\Phi(X) = \sum_a E_aXE_a^*$ for all $X \in B(H)$ that is ...
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Principal Symbol of the Fractional Laplacian on Manifolds
In the euclidean space $\mathbb{R}^n$, we can define the Fractional Laplacian as
$$(-\Delta)^s f := \int |\xi|^{2s}\hat{f}(\xi)e^{ix\cdot\xi}d\xi.$$
The principal symbol is clearly $p(x,\xi)=|\xi|^{2s}...
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Help understanding a proof about trace preserving and positive operators.
Edit: Going to try to make this better.
The implications (i) -> (ii) and (i) -> (iii) are clear to me. I'm not quite sure where the positive operators come from in (ii) -> (iii), but I had ...
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$T + \|T\| \cdot \bf 1$ is a positive operator?
I'm look at a proof where the author claims that we may assume that a self-adjoint operator $T$ is in fact positive by replacing $T$ with $T + \|T\| \cdot 1$...why can we do this?
Is $T + \|T\| \cdot \...
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