# 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.

9,626 questions
Filter by
Sorted by
Tagged with
10 views

### Bounding spectral norm of the matrix $f(\Sigma)$

Suppose that we have a sequence of matrices $\{\Sigma_n\in\mathbb{R}^{n\times n}\}_{n=1}^\infty$ with uniformly bounded spectral norm, that is to say, there exists a constant $C>0$, such that for ...
• 31
1 vote
24 views

### Solution to ODE in terms of eigenfunctions of differential operators

Can anyone please recommend a textbook or publication, where the theory of linear ordinary differential equations is explained from operator-theoretical perspective? That is, solutions to the ODEs are ...
• 11
18 views

### Spectral theorem for Normal Operators without using $C^*$ Algebras.

On page $263$ in Conway's book "A Course in Functional Analysis", the spectral theorem for normal operators is stated. It says that if $N$ is a normal operator, there is a unique spectral ...
• 69
20 views

• 941
46 views
+50

### Convergence of inverse operator with projections

Let $X$ be a separable Hilbert space, and let $(e_i)_{i=1}^\infty$ be an orthonormal basis of $X$. For each $n\in \mathbb{N}$, let $X_n$ be the subspace spanned by $(e_i)_{i=1}^n$, and consider the ...
• 12.6k
232 views

### Neumann series of the derivative operator

Consider the Nemuann series of the $D=\frac{d}{dx}$ operator: $T = 1 + D+ D^2+...$ Applying T on $x$ yields $x+1$ (i.e., $Tx=x+1$) for $x \in R$. I would appreciate if someone helps me to resolve the ...
• 284
51 views

### Bound on $\|ABv\|$ in terms of $\|Av\|$

I have a Hilbert space $H$ with: An unbounded self-adjoint operator $T$. Another unbounded operator $S$ such that $S=PTP^{-1}$, where $P$ is a bounded invertible operator (not necessarily unitary). ...
• 3,909
79 views

### Difference in usage between function, mapping, functional, form, and operator?

The word function has many synonyms (or close to synonyms), including: map functional form operator transformation What is the difference, in meaning or usage, between them? I understand that exact ...
• 2,131
1 vote
36 views

### If $P$ is self-adjoint and $P^2$ is a projection, when is $P$ a projection?

Let $H$ be a Hilbert space and $P:H \to H$ a linear operator. I am aware that if $P$ is self-adjoint and idempotent then it's a projection. My question is: if $P$ is self-adjoint and $P^2$ is a ...
31 views

### Show for $a$ in a C* algebra, there exists a unique element $b$ such that $bb^∗b = a$ [closed]

Let $A$ be a C*-algebra. The question is to show that for $a \in A$, there exists a $\textbf{unique}$ element $b\in A$ such that $bb^∗b = a$ . We prove existence like: Considering the polar ...
• 730
1 vote
47 views

### Idea about the projections of the $C^*$-algebra $pAp$, where $p$ is a projection in $A$

Let $A$ be a unital $C^*$-algebra and $p$ be a projection in $A$. Now consider the $C^*$-algebra $pAp$. I want to classify the projections in $pAp.$ I know that, if $q \in A$ is a projection such that ...
• 1,777
129 views

### Does the operator have any eigenvalues, also is it compact?

Question: Let $H=L^2([0,1],m)$, where $m$ is the Lebesgue measure, and consider th e operators $M,S\in L(H,H)$ given by $$Mf(t)=tf(t), \; Sf(t)=f(1-t), f\in H, t\in [0,1]$$ You are not asked to ...
• 33
27 views

• 11
28 views

• 12.6k
27 views

1 vote
17 views

### When and where the Laplacian operator is bounded?

aIn what domains and spaces is the Laplacian operator closed? If $D(\Delta):=\left\{u\in L^2:\Delta u^2\ \text{ in } L^2\right\}$ (maximal domain) Question 1. Is $\Delta:D(\Delta)\subset L^2\to L^2$ ...
• 2,880
16 views

### What is the trace of the second tensor component?

My teacher asks to prove that for any matrix $A \in \operatorname{Mat}(N, \mathbb{C})$ there is true: $$A=\operatorname{tr}_2\left(P_{12} A_2\right)$$ where $A_{2}=E_{N}\otimes A$ and $P_{12}$ is ...
• 23
1 vote
38 views

• 513
48 views

### Is this Hardy-like operator $f \mapsto x^{-\frac{1}{2}} \int_{x}^{2x} f(y) dy$ bounded from $L^2$ to $L^2$?

I'm interested in knowing whether $$Tf(x) = x^{-\frac{1}{2}}\int_{x}^{2x}f(y)dy = \int_{\mathbb R} K(x,y)f(y)dy, \quad x > 0$$ where $K(x,y) = x^{-\frac{1}{2}} 1_{0 < y \leq x \leq 2y}(x,y)$, ...
• 1,875
1 vote
29 views

### How to show that an operator in $B(L^2(N))$ is actaully in $N$?

I am reading a note written by Claire Anantharaman and Sorin Popa recently. Here is a link to the notes. https://www.math.ucla.edu/~popa/Books/IIun.pdf I am reading Chapter 13 right now, in section 13....
• 368
62 views

### Does $x^T T(y) = y^T T(x)$ imply that $T$ is a linear operator?

Let $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ an operator satysfying: \begin{equation} x^T T(y) = y^T T(x) ~~~~\forall (x,y)\in \mathbb{R}^n\times \mathbb{R}^n. \end{equation} Does it imply that $T$ ...
• 87
1 vote
23 views

### Norm and conjugate of operator on $l^p$

I want to solve the following problem: Show that for every sequence $(\alpha_n) \in l^{\infty}$ formula $$A(x_n) = (\alpha_n x_n), (x_n) \in l^p,$$ gives bounded linear operator on $l^p$, find its ...
• 161
1 vote
20 views

20 views

### Dual operator of Markovian operator

I am trying to incorporate the definition of dual operator in a Markovian setting. Say I have a Markov kernel $K:S\times S\to[0,1]$ i.e., a mapping such that for every $A\in S$ $x\to K(x,A)$ is a ...
• 889
1 vote
46 views

### Can we have some examples about i.c.c groups?

In the following discussion, we always assume $G$ is a countable discrete group. I am learning about the group von Neumann algebra recently, We know that $L(G)$ is a factor if and only if $G$ is an i....
• 368
32 views

### Finding norm of integral operator on C[0,1] [duplicate]

I need help with the following: Find norm of inegral operator $K:C[0,1] \to C[0,1]$ which is defined as $$Kf(x) = \int_0^1 k(x,y)f(y)dy, \forall f \in C[0,1], x \in [0,1],$$ where $k \in C[0,1]^2$. ...
• 161
1 vote
31 views

### An identity of the resolvent operator of the Laplace operator

Let $\Omega \subset \mathbb{R}^n$ be an open bounded set. Consider the following operator $R=(\lambda I-\Delta)^{-1},$ where $\lambda>0$ and $I$ denotes the identity operator. Then, by the Hille-...
• 139
1 vote
31 views

### Direct integral

Let us assume that every w*-continuous bounded unital homomorphism from a separably acting type II_1 factor $M$ to some $\mathbb B(H),$ where $H$ is a Hilbert space, is completely bounded. $(1)$ Let ...
31 views

### All spectral projection is trivial implies the operator is identity?

We consider $x\in B(H)$ a self-adjoint operator, then we know that it can determine a unique spectral measure $E$. If we know that $E(\Delta)=0$ or $1$ for all measurable $\Delta\subset \sigma(x)$, ...
• 368
1 vote
22 views

### A question about standard representation for type $II_1$ factors?

We suppose that $M$ is a type $II_1$ factors, then we have a standard representation, we know that every element in $L^2(M)$ can be approximated by elements in $M$ by 2-norm, in another word, for ...
• 368
39 views

### If $A \in \mathcal{B}(H), B \in \mathcal{B}(K)$ are contractions, is it then true, that $\begin{bmatrix}A&0\\0& B \end{bmatrix}$ is a contraction too?

$A \in \mathcal{B}(H)$ is a contraction, if: $$\| A x \| \le \| x \|$$ for every $x \in H$. In other words: $\| A \| \le 1$. The question here is: How can I show, that a matrix of operators (which ...
• 513
24 views

### Nets and closedness of $\mathcal{B}(H)$ in the topology of pointwise convergence

Let $H$ be an infinite-dimensional Hilbert space and $\mathcal{B}(H)$ the set of bounded linear operators on $H$. One way to define the strong operator topology (SOT) on $\mathcal{B}(H)$ is by ...
• 4,542
32 views

### Boundedness of integral operator induced by kernel $K(x,y) := \frac{1}{x+y}$.

Let $t_0 > 0$, $p \in (1,+\infty)$ and define $K:(0,t_0)\times (0,t_0) \rightarrow \mathbb{R}$ by $K(x,y) := \frac{1}{x+y}$. Is it true that for all $f \in L^p\bigl((0,t_0);\mathbb{R}\bigr)$ the ...
52 views

### Is $A\in B\left( \overline{H}\right)$ invertible if it is onto and $\left. A\right\vert _{H}\in B\left( H\right)$ is one-to-one and onto?

Let $\overline{H}$ be a Hilbert space and $A\in B\left( \overline{H}\right)$ be a surjective operator such that $A\left( H\right) =H$ and its restriction to $H$ is injective and. I'm asking if under ...
• 1,001
1 vote
### Extension of $\{f\in C([0, 1], B)\,\vert\, f(0)=f(1)=0\}$ by $A$ with $\ast$-homomorphism $\phi:A\rightarrow B$
The following question is from An Introduction to $K$-theory for $C^{\ast}$-Algebra and an e-copy can be found here. Below is the question (since I do not know how to create a diagram in MS ...) By ...