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
351
votes
0
answers
21k
views
Limit of sequence of growing matrices [closed]
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(\...
- 7,735
208
votes
7
answers
16k
views
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 ...
- 26.3k
85
votes
8
answers
26k
views
Equivalent Definitions of the Operator Norm
How do you prove that these four definitions of the operator norm are equivalent?
$$\begin{align*}
\lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for ...
- 921
80
votes
4
answers
3k
views
Factorial of a matrix: what could be the use of it?
Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
- 933
60
votes
11
answers
31k
views
What is an operator in mathematics?
Could someone please explain the mathematical difference between an operator (not in the programming sense) and a function? Is an operator a function?
- 803
51
votes
2
answers
25k
views
What is the difference between isometric and unitary operators on a Hilbert space?
It seems that both isometric and unitary operators on a Hilbert space have the following property:
$U^*U = I$ ($U$ is an operator and $I$ is an identity operator, $^*$ is a binary operation.)
What ...
- 8,335
50
votes
4
answers
4k
views
Double sum - Miklos Schweitzer 2010
There is a question in the Miklos Schweitzer contest last year that keeps bugging me. Here it is:
Is there any sequence $(a_n)$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 &...
- 23.5k
44
votes
4
answers
14k
views
Differential equations and Fourier and Laplace transforms
Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
- 3,235
41
votes
1
answer
1k
views
Coordinate free proof that $\operatorname{trace}(A)=0\:\Longrightarrow\:A=BC-CB$
As you probably know, the trace function on square matrices has the property that
$$\operatorname{trace}(AB-BA)=0\,.$$
You might also know that the converse is true:
$$\operatorname{trace}(A)=0\;\text{...
- 15k
41
votes
0
answers
780
views
$A$ 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 $...
- 11.8k
40
votes
4
answers
37k
views
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 ...
- 1,536
38
votes
1
answer
642
views
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 ...
- 15.3k
37
votes
4
answers
22k
views
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 ...
- 3,350
32
votes
1
answer
751
views
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 ...
- 423
30
votes
3
answers
15k
views
The difference between hermitian, symmetric and self adjoint operators.
I am struggling with the concept of hermitian operators, symmetric operators and self adjoint operators. All of the relevant material seems quite self contradictory, and the only notes I have do not ...
- 11.9k