Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Operator theory is the study of linear operators between various (topological) vector spaces. The quintessential examples being differential and integral operators on function spaces over some domain, and hence operator theory is deeply connected to functional analysis. It has applications to and applications from differential equations, representation theory and mathematical physics.

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Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ $K_1=\left(\begin{array}{c}1 \\ 0\end{array}\right)$ and consider the sequence of matrices defined by $$ K_L =…
Eckhard
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How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any sense, with $$\mathsf{A} = \left(\begin{matrix} 1…
Laplacian
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Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma function. What use might it be to take the factorial of…
Oliphaunt
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Equivalent Definitions of the Operator Norm

How do you prove that these four definitions of the operator norm are equivalent? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ &=\sup\{ \lVert Av\rVert\;\colon\;…
KiaSure
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What is an operator in mathematics?

Could someone please explain the mathematical difference between an operator (not in the programming sense) and a function? Is an operator a function?
Nik
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Double sum - Miklos Schweitzer 2010

There is a question in the Miklos Schweitzer contest last year that keeps bugging me. Here it is: Is there any sequence $(a_n)$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 <\infty $ and $$\sum_{n \geq 1}\left(\sum_{k \geq…
Beni Bogosel
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Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and some other situations where the Laplace transform…
user782220
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How to justify solving $f(x+1) + f(x) = g(x)$ using this spectral-like method?

Let's say that I want to find solutions $f\in C(\Bbb R)$ to the equation $$ f(x+1) + f(x) = g(x) $$ for some $g\in C(\Bbb R)$. I can write $f(x+1) = (Tf)(x)$ where $T$ is the right shift operator and rewrite the equation suggestively as $$ (I+…
BigbearZzz
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What is the difference between isometric and unitary operators on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is an identity operator, $^*$ is a binary operation.) What is the difference between isometry and unitary?…
sleeve chen
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Norm of a symmetric matrix equals spectral radius

How do I prove that the norm of a matrix equals the absolutely largest eigenvalue of the matrix? This is the precise question: Let $A$ be a symmetric $n \times n$ matrix. Consider $A$ as an operator in $\mathbb{R}^n$ given by $x \mapsto Ax$. Prove…
Erik Vesterlund
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$A$ and $B$ commute on a dense set but $e^{iA}$ and $e^{iB}$ do not

Let $A$ and $B$ be unbounded, symmetric operators on a Hilbert space $H$ with a common domain $D$. If $AB = BA$ on $D$, is it necessarily that case that $e^{iA}$ and $e^{iB}$ also commute? If $A$ and $B$ are bounded, then I know that this must be…
user15464
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Does $\sigma(T) = \{1\}$ and $\|T\| = 1$ imply that $T$ is the identity?

Suppose that $T$ is a bounded linear operator on a complex Banach space X and that we know that $\sigma(T) = \{1\}$ and $\|T\| = 1$ (i.e. the spectrum of the contraction $T$ consists only of a single point, 1). Does it follow that $T$ is the…
Markus Hasenöhrl
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Is it possible to characterize completeness of a normed vector space by convergence of Neumann series?

If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? Context: It is well known that if $X$ is a Banach space and…
PhoemueX
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Why do zeta regularization and path integrals agree on functional determinants?

When looking up the functional determinant on Wikipedia, a reader is treated to two possible definitions of the functional determinant, and their agreement is trivial in finite dimensions. The first definition is based on zeta function…
anon
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Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$?

I am a beginner of funcional analysis. I have a simple question when I study this subject. Let $L(X)$ denote the Banach algebra of all bounded linear operators on Banach space X, $T\in X$ is invertible, then $||T^{-1}||=||T||^{-1}$? Is this result…
Yan kai
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