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.

6,346 questions
11k views

22k views

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?
13k views

867 views

Selfadjoint compact operator with finite trace

I have a compact selfadjoint operator $T$ on a separable Hilbert space. For some fixed orthonormal basis, the operator's diagonal is in $\ell^1(\mathbb{N})$. Can we conclude that $T$ is trace ...
1k views

Find the spectrum of the linear operator $T: \ell^2 \to \ell^2$ defined by $Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{Z}}$

Let $\ell^2 =\ell^2(\mathbb{Z})$. Choose $\theta \in ]0,1[$ and set: $$Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{Z}}$$ for each $x=(x_n)_{n\in \mathbb{Z}}\in \ell^2$ (thus $T$ is a convex ...
1k views

What are some meaningful connections between the minimal polynomial and other concepts in linear algebra?

I’ve found that the most effective way for me to deeply grasp mathematical concepts is to connect them to as many other concepts as I can. Unfortunately, I’m seeing neither the importance nor the ...
746 views

Consider the left shift operator $T : \ell^1(\mathbb N) \to \ell^1(\mathbb N)$ by $$T(x_1,x_2..... )=(x_2, x_3 ........),$$ and also the right shift operator $S : \ell^1(\mathbb N) \to \ell^1(\mathbb ... 2answers 2k views Closure of the invertible operators on a Banach space Let$E$be a Banach space,$\mathcal B(E)$the Banach space of linear bounded operators and$\mathcal I$the set of all invertible linear bounded operators from$E$to$E$. We know that$\mathcal I$... 1answer 521 views Injectivity of the operator$(Ax)(t)=\int_0 ^1 k(s,t) x(s)ds$Let$X=C([0,1],\mathbb{R})$(equipped with the supremum norm). Let$A$be the operator defined for each$x\in X$by $$(Ax)(t)=\int_0 ^1 k(s,t) x(s)ds,$$ where$k:[0,1]\times [0,1]\to \mathbb{R} $is ... 0answers 1k views Why is the numerical range of an operator convex? Let$T$be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that$W(T)$is a convex subset of the complex ... 3answers 14k views An inequality on trace of product of two matrices Suppose we have two$n \times n$positive semidefinite matrices,$A$and$B$, such that$\mbox{tr}(A), \mbox{tr}(B) \le 1$. Can we say anything about$\mbox{tr}(AB)$? Is$\mbox{tr}(AB) \le 1 $too? 1answer 404 views How does$\sigma(T)$change with respect to$T$? Consider$\sigma$as a mapping which maps$T\in\mathcal{L}(X)$to$\sigma(T)$, the spectrum of$T$, a compact set in the complex plane. I wonder whether there is some result concerning how$\sigma(T)$... 1answer 489 views Is this a characterization of commutative$C^{*}$-algebras Assume that$A$is a$C^{*}$-algebra such that$\forall a,b \in A, ab=0 \iff ba=0$. Is$A$necessarily a commutative algebra? In particular does "$\forall a,b \in A, ab=0 \iff ba=0$" ... 1answer 510 views Singular-value inequalities This is my question: Is the following statement true ? Let$H$be a real or complex Hilbertspace and$R,S:H \to H$compact operators. For every$n\in\mathbb{N}$the following inequality holds: $$\... 1answer 932 views Looking for an easy lightning introduction to Hilbert spaces and Banach spaces I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to participate,... 3answers 2k views Is there a formula similar to f(x+a) = e^{a\frac{d}{dx}}f(x) to express f(\alpha\cdot x)? Using the Taylor expansion$$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\... 3answers 7k views 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 ... 3answers 1k views 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\...
Hoi, consider the Hilbertspace $l^2$ and the Left and Right-shift operator \begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*} I know ...