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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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 ...
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Definition of resolvent set
I'm having trouble understanding some subtlety of definition of resolvent set for given bounded operator A everywhere defined on some Hilbert space. Book I use (and many other sources) give the ...
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Show that $P_i$ and $\sum_i P_i$ being idempotent implies $P_i P_j=\delta_{ij}$
Let $X$ be a finite dimensional real linear space, or more generally a finite dimensional vector space over a field of characteristic $0$.
Let $(P_i)_{i=1}^n$ be a finite sequence of linear mappings $...
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Exponential of a function times derivative
Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e.
\begin{equation}
e^{a\partial}f(x)=f(a+x)
\end{equation}
This can be easily verified from a Taylor series
\begin{equation}
...
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Question about Angle-Preserving Operators
This an exercise out of Spivak's "Calculus on Manifolds".
Edit: There was a typo in the exercise as is noted below in the answers. The statement has been edited to reflect this.
Given $x,y\in\...
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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 ...
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Easy Proof Adjoint(Compact)=Compact
I am looking for an easy proof that the adjoint of a compact operator on a Hilbert space is again compact. This makes the big characterization theorem for compact operators (i.e. compact iff image of ...
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Is there a such thing as an operator of operators in mathematics?
Thus far I have seen operators of numbers and operators that perform on functions like Laplace, Fourier and Z-Transforms but is there an operator in existence that performs on other operators?
Like a ...
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Norm of self-adjoint operator
I am trying to prove that $\|A\|=\sup_{\|x\|=1}|\langle x,Ax\rangle|$ for some self-adjoint bounded operator $A$ on a Hilbert space.
Can anyone give me a hint how to prove it.
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How to prove that an operator is compact?
Consider $T\colon\ell^2\to\ell^2$ an operator such that
$Te_k=\lambda_k e_k$ with $\lambda_k\to 0$ as $k \to \infty$ how to prove that it is compact?
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Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?
Using the Taylor expansion
$$f(x+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dx^k}f(x)$$
one can formally express the sum as the linear operator $e^{a\frac{d}{dx}}$ to obtain
$$f(x+a) = e^{a\...
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Norm of integral operator in $L_2$
What is the norm of integral operators $A$ in $L_2(0,1)$?
$Ax(t)=\int_0^tx(s)ds$
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Compact operators on an infinite dimensional Banach space cannot be surjective
I am reading a book about functional analysis and have a question:
Let $X$ be a infinite-dimensional Banach-space and $A:X \rightarrow X$ a compact operator. How can one show that $A$ can not be ...
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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 ...
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Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$?
I am a beginner of functional 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 ...
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