Tagged Questions

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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Exercise 34 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 34 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 201): Let K be a Hilbert-Schmidt kernel which is real and symmetric. Then, as we saw, the operator $T$ whose ...
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Problem 8 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein and Shakarchi's Real Analysis

The following is problem 8 from chapter 4 ("Hilbert Spaces: An Introduction") of Stein and Shakarchi's Real Analysis. Suppose $\{t_k\}$ is a collection of bounded operators on a Hilbert space $H$. ...
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Prove that T is compact

If $H$ is a Hilbert space with basis $\{\varphi_{k}\}^{\infty}_{k=1}$, how do I show that the operator $T$ defined by $T(\varphi_{k})=\frac{1}{k}\varphi_{k+1}$ is compact and has no eigenvectors? ...
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Compact Operator on Hilbert Space

How do I show that the range of $\lambda I-T$ is all of $H$ (Hilbert Space) if and only if the null-space $\bar\lambda I-T^{\ast}$ is trivial? Thanks!
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If $T$ is topologically transitive and $X$ is separable and complete then there exists a dense set of points with dense backward orbits.

I am trying to solve exercise 1.2.7 from Grosse-Erdmann and Peris' book Linear Chaos. It is stated as follows: Let $T:X\rightarrow X$ be continuous on a separable and complete metric space $X$ ...
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A matrix $G$ with all eigenvalues with nonzero real part. Then $t\mapsto |\exp(tG)x |$ is unbounded

I am trying to see why this is true. A book I am reading has this claim without any verification and I'm trying to see why it is true. Let $G$ be an $n\times n$ matrix all of whose eigenvalues have ...
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What is the closest self-adjoint (positive) operator to a given operator?

Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
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Riesz Functional Calculus vs. Holomorphic Functional Calculus

"Functional calculus" is a word used to describe the practice of taking some functions or formulas defined on complex numbers, and apply them in some way to certain kinds of operators, despite that ...
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Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*)$...
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Question about the Averson's proof of the bicommutant theorem.

In the Averson's proof of the bicommutant theorem is proved that, if $A$ is a self-adjoint algebra of operators with trivial null space and $T \in A''$, for every $\epsilon>0$, $n=1,2..$ and every ...
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Lipschitz constants of projections

Consider two compact sets $A, B \subset \mathbb{R}^n$. Assume that the projection mappings $P_A: \mathbb{R}^n \rightarrow A$, $P_B : \mathbb{R}^n \rightarrow B$ have Lipschitz constant $1$ and $L$, ...
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Let $A \colon \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ be the linear operator defined by $\left( Ax \right)_k = \sum_{i \in \mathbb{Z}}a_{ki}x_i$, where $a_{ki} = 1/(k-i)^2$ if $k \neq i$ and $a_{... 1answer 90 views Proof of operator norm with the equality of involution Given the equality $$\|A^*A\| = \sup_{\| x \| = \| y \| = 1} | (Ax, (A^*)^*y) |$$ How do show that it is equal to$\|A\|^2$Is it by using Cauchy-Schwarz inequality such that$(x, y) \leq \|x\|\|y\...
If $T$ is an differential operator of order $k$ from $\mathbb{C}$-vector bundle $E$ to $F$ over a compact differential manifold $X$. Question: how can we extend it to a continuous linear map between ...
Existence of a semigroup of bounded operators which is not $C_0$
Let $X$ be any Banach space. Then we can define a $C_0$ semi group of bounded operators on $X$. But my question is that can we define a semi group of bounded operators which is not $C_0$?