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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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 $(...
