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.
10,436
questions
1
vote
3
answers
608
views
what is/are the spectrum of operators and their applications
this is an educational question.
can someone please explain with some simple examples:
(1) what is/are the spectrum of operator
(2) where it is useful
For providing examples of spectrum of ...
- 691
17
votes
1
answer
2k
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 ...
- 6,291
7
votes
2
answers
967
views
On the isometry between bounded linear operators and the dual of nuclear linear operators
Let $H$ be a separable Hilbert space. Let $(e_i)_i$ be an orthonormal basis. For any bounded linear map $T$ we write, whenever possible
$$\operatorname{tr} T := \sum_{i}^{\infty} \langle T e_i, e_i \...
- 7,670
8
votes
1
answer
1k
views
Uniform mean ergodic theorem
I'm working in Einsiedler and Ward's book on Ergodic Theory and in Exercise 2.5.4 they want to prove the following
$$\lim_{N - M \to \infty} \frac{1}{N - M} \sum_{n = M}^{N - 1} U_T^n f \to P_T f.$$
...
- 14.7k
0
votes
1
answer
484
views
Linear operator categories
Let's consider linear operators on the set of complex-valued functions to the same set. I wonder to which categories such operators can be classified. All linear operators I encountered so far fall ...
- 9,261
2
votes
3
answers
995
views
Scale Operator $Uf(x)=f(kx)$
I am looking for an operator $U$, that can do this to a function:
$$Uf(x)=f(2x).$$
In particular I am happy if there is an $U$ for the general case:
$Uf(x)=f(kx)$.
Does such an operator exist for ...
- 8,759
7
votes
2
answers
557
views
Finding all linear operators $L: C([0,1]) \to C([0,1])$ which satisfy 2 conditions
As above, I'm trying to find all linear operators $L: C([0,1]) \to C([0,1])$ which satisfy the following 2 conditions:
I) $Lf \, \geq \, 0$ for all non-negative $f\in C([0,1])$.
II) $Lf = f$ for $f(...
Stephen Gregg
2
votes
1
answer
3k
views
Cross product of operators
How to show that:
$ (-i\nabla-eA)\times(-i\nabla-eA) = (ie\nabla \times A) $
i and e are constants
A is a vector field
$\nabla$ = vector differential operator
- 8,759
16
votes
2
answers
6k
views
What is operator calculus?
I watched the excellent interview with Richard Feynman: http://www.youtube.com/watch?v=PsgBtOVzHKI In the interview Feynman mention that he at young age re-invented operator calculus.
I have searched ...
- 263
5
votes
2
answers
559
views
If $(I-T)^{-1}$ exists, can it always be written in a series representation?
If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm < $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$.
Thinking in terms of a ...
- 8,633
9
votes
1
answer
2k
views
Identities with Div, Grad, Curl
In physics there are lots of identities like:
$$\nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - (\nabla \cdot \nabla) A$$
I'm wondering if there is an algorithmic algebraic method to ...
anon