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.

learn more… | top users | synonyms

4
votes
1answer
51 views

Idempotents which are not Murray-von Neumann equivalent to its adjoint

What is an example of a $C^{*}$ algebra with an idempotent $e$ such that $e$ is not Murray-von Neumann equivalent to $e^{*}$?
0
votes
1answer
10 views

How does the general spectral theorem generalize the simpler versions?

I read the following version of the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I'm trying to understand why this is a generalization of the following version, ...
1
vote
0answers
8 views

$V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 : the semi group $V(A)$ of equivalent projections (under Murray Von Neumann equivalence) in $M_∞(A)$ is ...
4
votes
3answers
105 views

How do I get $\|x\|\le C\|y\|$ in this case?

I feel that the title is a bit uninformative, please feel free to edit it. This is a problem related to the Open Mapping Theorem. Let $T:X\to Y$ be a bounded linear operator from a Banach space X to ...
0
votes
1answer
17 views

Reference request: Every vector subspace of codimension $1$ of a vector space $A$, is the kernel of some nonzero functional $f$.

I know the following statement is true, but I am looking to find a good reference that proves this quite nicely Every vector subspace of codimension $1$ of a vector space $A$, is the kernel of ...
6
votes
2answers
561 views

Double dual of the space $C[0,1]$

The second dual or double dual of the space of all continuous functions on $[0,1]$, $C[0,1]$ is von Neumann algebra. Can anyone help me identifying this space?
1
vote
0answers
8 views

Difference between Schmidt decomposition and singular value decomposition

Schmidt decomposition of an operator is a useful tool of quantum information theory nowadays. Let $O$ be an operator acting on the Hilbart space $\mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}$. ...
1
vote
3answers
34 views

Examples of bounded linear operators with range not closed

I've been trying to get some intuition on what it means for a bounded linear operator to have closed range. Can anyone give some simple examples of such an operator that does not have closed range? ...
0
votes
1answer
37 views

Regarding integral operators being contractions

I have two half-questions that tie into one another. Suppose $T$ is an operator on $C([0, 1])$ defined by $$(Tu)(t) = \int_0^t (u(x))^2dx.$$ Show that T is not a contraction on the closed unit ball ...
0
votes
1answer
73 views

$e^{(A+B)} = e^Ae^Be^{[A,B]}$ for non commuting A and B?

For non commuting A and B, and the derivative of $[A,B] = 0$. Is it true that/how to prove that $e^{(A+B)} = e^Ae^Be^{[A,B]}$ If not, what is the expression according to Wikipedia's article on the ...
1
vote
1answer
49 views

support of an operator on a Hilbert space

Let $x\colon\mathcal{H}\to\mathcal{H}$ be a self-adjoint operator, the support $s(x)$ of $x$ is defined as the smallest projection $e\in B(\mathcal{H})$ such that $ex=xe=x$. Let $x=\int\lambda \, ...
3
votes
1answer
75 views

Closure of an Operator in $l^2$

Let $l^2$ denote the Hilbert space of all complex sequences $\phi = (\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j|^2 < \infty$. Consider the linear subspace of $l^2$ defined by ...
102
votes
5answers
4k 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 ...
2
votes
1answer
37 views

A relation between two properties of sequences of operators

We have $(T_l)_l$ a sequence of bounded linear operators from $\ell^2$ to $\ell^2$. $\bullet$ We say $(T_l)_l$ satisfies the property "A" if ...
2
votes
1answer
64 views

Does $AB=(AB)^{\ast}$ and $A=A^{\ast}$ implies $B=B^{\ast}$?

Suppose that we have $AB=(AB)^{\ast}$ and $A=A^{\ast}$, does this implies that $B=B^{\ast}$? ($A^{\ast}$ is the Hermitian adjoint of $A$.) I have a feeling that they might not be equal in general. ...
1
vote
0answers
26 views
+50

Finite dimensional Banach algebras whose $K_{0}$ group is a non trivial finite group

Motivated by this question we ask Is there a finite dimensional Banach algebra $A$ such that $K_{0}(A)$ is a nontrivial finite group? I understand from the above link and this post that any ...
0
votes
0answers
19 views

Norm convergence of a net of operators

Let $T$ be a positive operator in B(H). For every $\epsilon >0$, define $T_{\epsilon}:=(T+\epsilon I)^{-\frac{1}{2}}$. This makes sense since the spectrum of $T$ lies in $[\epsilon,\infty)$. Let ...
3
votes
2answers
47 views

Intuition behind: Integral operator as generalization of matrix multiplication

So I am teaching myself more in-depth about integral operators and every once and awhile I see this little 'factoid', that integral operators are generalizations of matrix multiplications. In ...
3
votes
1answer
31 views

Algebra of Linear differential operators, question on Commutativity and Association

The following is a discussion on the following second differential equation $$ \frac{dy^2}{dx} - y = 0 $$ So, let us introduce the following, convention and definition, represent the derivative ...
1
vote
1answer
23 views

the c*-algebra generated by the Volterra operator

Let V be the Volterra operator on $\mathscr{L^2(0,1)}$.$V(f)(x)=\int_{0}^{x}{f(y)dy}$. Show that $C^*(V)$, the smallest C* algebra generated with V, is $\mathbb{C}+\mathscr{B_0(L^2(0,1))}$ where ...
0
votes
0answers
24 views

Distance from image of bounded operator

Let $A:H\rightarrow H$ be a bounded linear operator on Hilbert space $H$. Suppose we have $x\in H$ and $r>\mathrm{dist}(x,A(H))=\inf\{\|x-Ah\|,\ h\in H\}$. How to prove that then there exist ...
0
votes
3answers
36 views

Spectral radius and dense subspace

Let $X$ be a Banach space, and let $E$ be a dense subspace of $X$. Let $A: X \to X$ be a bounded operator on $X$ that maps $E$ to itself. Assume that the spectral radius of $A$ restricted to $E$ is ...
3
votes
2answers
34 views

Composition involving bounded linear operators

I recently come across the following statement mentioned in a proof: Let $X,Y$ be normed linear spaces and $T:X \rightarrow Y$ be a linear operator. if for every bounded linear functional $U: Y ...
19
votes
3answers
5k 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 ...
2
votes
0answers
106 views

Complex Root of Unity Analogue of Forward Difference Operator

In my studies I have come across a couple of operators; in particular; $$\Delta[f(x)]=f(x+1)-f(x)$$ $$\Delta^*[f(x)]=f(x+1)+f(x)$$ $\Delta$ has been called the Forward Difference Operator. I was ...
0
votes
1answer
14 views

infimum of operator norms of iterations of linear operators

I am currently reading a proof in which a fact is used without proof: For a Banach space $X$ and a bounded linear operator $T: X \to X$, $$ \lim_{n \to \infty} \| T^n \|^{\frac{1}{n}} = \inf_{n ...
0
votes
0answers
35 views

example of positive operators a,b, $a\le b$ but $b^2-a^2$ is not positive [closed]

Give an example of a C*-algebra $\mathscr{A}$ and positive elements a,b in $\mathscr{A}$ such that $a\le b$ but $b^2-a^2\notin \mathscr{A}_+$, i.e. $b^2-a^2$ is not positive element in $\mathscr{A} $
13
votes
0answers
241 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 ...
2
votes
2answers
40 views

Matrix of linear operator in different bases [PMA Rudin]

$\mathbf{9.35}\quad$ Theorem $\ \; $ If $[A]$ and $[B]$ are $n$ by $n$ matrices, then $$\det([B][A])=\det[B]\det[A].$$ $\mathbf{9.36}\quad$ Theorem $\ \; $ A linear operator $A$ on $R^n$ is ...
0
votes
0answers
18 views

the c*-algebra generated by a closed ideal and a c*-subalgebra

If $\mathscr{A}$ is an unital c*-algebra, $I$ is a closed ideal of $\mathscr{A}$ ,and $\mathscr{B}$ is a unital c*-subalgebra of $\mathscr{A}$ . Show that the c*-algebra generated by ...
1
vote
1answer
27 views

Support of a normal pure state is a rank one projection

Let $\phi$ be a normal pure state on a w*-algebra $M$ and $\{\pi, \xi, H\}$ its GNS representation associated to $\phi$. Suppose projection $e$ is the support of $\phi$. Show that $\pi(e)$ is a rank ...
1
vote
0answers
30 views

Convolution operator is normal

Consider the convolution operator $$Tf(s)=\frac{1}{2\pi}\int_0^{2\pi}f(t)h(s-t)\,\,dt,\quad f\in L^2[0,2\pi]$$ where $h:\Bbb R\to \Bbb C$ is a $2\pi$-periodic function, square integrable on ...
1
vote
1answer
251 views

Comparison of Strong OPerator and Weak * Topologies on B(H)

It is known that in $\mathfrak{B}(\mathbb{H})$, the weak operator topology (WOT) is contained in both the strong operator topology (SOT) and $\sigma$-weak topology. In general the SOT and the ...
5
votes
1answer
53 views

Continuous but not compact operator on $L^2(0,\infty)$

Define the following operator on $L^2(0,\infty)$: $$Tf(x)=\frac{1}{x} \int_0^xf(y)dy,\quad f\in L^2(0\infty).$$ I would like to see that it is continuous but not compact. So, this is an integral ...
0
votes
2answers
19 views

Self-adjoint operators, projections, and resolutions of the identity.

In my Functional Analysis course, we're discussing the Spectral Theorem and the like. One question from a previous exam states the following: Let $H$ be Hilbert over $\mathbb C$, let $T \in B(H)$ be ...
1
vote
1answer
23 views

Closed ideals in $\mathbb B(H)$

Let $\mathbb{H}$ be a non-separable Hilbert space. If $\alpha$ is an countably many infinite cardinal number, let $I_{\alpha}=\{A\in \mathbb{B(H)}\:dim~ cl(ran A)\le \alpha\}.$ Show that ...
2
votes
1answer
18 views

Positive self-adjoint operators and norm resolvent convergence

I recently came across a reference to the following Theorem (Simon/Reed, Methods of Modern Mathematical Physics, viii.25) and am now trying to figure out a proof for it: If $A_n$ and $A$ are positive ...
4
votes
1answer
51 views

Example of operator with spectrum equal to $\mathbb{C}$?

In my Functional Analysis course, we proved that for a (possibly unbounded) operator $T$ that is densely defined, closed, and symmetric, exactly one of the following four occurs: $\sigma(T) = ...
1
vote
2answers
484 views

Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$ A_f(g) = \int f(\tau)g(t-\tau)d\tau $$ and apply it to $g(t)=e^{ikt}$. ...
3
votes
1answer
44 views

Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$ Some definitions and results of the ...
1
vote
0answers
21 views

How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$?

Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha ...
3
votes
1answer
29 views

‎Jointly ‎continuous of product in $B(H)$

‎Let ‎$‎B(H)‎$ ‎be the set of‎ ‎bounded ‎operators ‎on a Hilbert space ‎‎$‎‎H$.‎ ‎ I ‎know that ‎$u_{‎\alpha‎}‎\longrightarrow u‎$‎‎‎ ‎in ‎S.O.T ‎if ‎and ‎only ‎if‎ ‎$u_{‎\alpha‎}(x)‎\longrightarrow ...
0
votes
2answers
58 views

Adjoint of an Operator in $l^2$

Let $l^2$ be the Hilbert space of all complex sequences $\phi =(\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j |^2 < \infty$. Set $D= \{ \phi \in l^2 : \sum_{j=0}^{\infty} j ...
1
vote
1answer
20 views

Largest invariant subspace

If $A$ is a $n\times n$ matrix with complex entries, denote by $inv(A)$ the dimension of a largest dimensional non-trivial invariant subspace of $A$. What is: $$\inf\{inv(A): ...
2
votes
1answer
20 views

Small perturbations and eigenvalues

Suppose $A$ is a $n\times n$ matrix. Given $\epsilon>0$, can one find a rank one matrix $B$ with euclidean norm at most $\epsilon$ such that $A+B$ has $n$ distinct complex eigenvalues? Given a ...
0
votes
1answer
30 views

Double dual of the space of bounded operators on Hilbert space [duplicate]

Every Banach space $X$ is canonically, isometrically embedded in its bidual $X^{**}$. But it is not always $1$-complemented in the bidual: for example, there is no projection from $\ell_\infty$ onto ...
1
vote
0answers
44 views

Eigenfunctions of integral operator

I am faced with the problem of calculating the eigenfunctions for an operator of the form: $(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy $ Does anyone know for which functions (or types of ...
2
votes
2answers
50 views

Prove that there is no norm for to make this mapping continuous

I am dealing with an exercise which is as follows: Show that there is no norm such that the set of all the mappings $T_a$ which map every element $f\in C(\mathbb{R}, \mathbb{R})$ (where the latter is ...
7
votes
4answers
990 views

Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels

I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of ...
2
votes
1answer
42 views

Square Root of the shift operator indexed by $\mathbb{Z}$

My question is very similar to this question, but instead of indexing by $\mathbb N$ I am indexing by $\mathbb Z$. Consider the shift operator $T : \ell^1(\mathbb Z) \to \ell^1(\mathbb Z) $ given by ...