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
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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 ...
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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
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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 ...
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Analitycity of band functions
Let $H=-\dfrac{d^2}{dx^2}+V(x)$, where $V(x)$ is a $2\pi$-periodic and $V\in L^{\infty}(\mathbb R)$.
If $\mathcal H':=L^{2}(0,2\pi)$, we have the decomposition
$\mathcal H=\int_{\mathbb [0,2\pi)}^{\...
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Can non-surjective bounded and bounded-below operators approach an invertible operator as the limit?
Let $\mathcal{H}$ be a Hilbert space. If $\left(A_n\right)$ is a norm-convergent sequence of bounded, bounded-below, but non-surjective operators such that the limit operator $A$ is also bounded and ...
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1
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Compactness of Differential Operator between Sobolev and $L^p$-spaces
I was wondering under which conditions the (weak) differential operator $D: W^{k,p}(\Omega)\rightarrow L^p(\Omega)$, $u \mapsto Du$ from a Sobolev space into the underlying $L^p$-space (on some open ...
- 41
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Understanding when two polynomials have the same linear operator (Dunford and Schwartz)
Hello. I'm trying to understand the first part of this theorem (theorem 3 in linear operators by Dunford and Schwartz) by writing it my way but I can't quite understand some steps. Next, I will put ...
- 2,880
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If $f\in I$, then $f^*=\overline{f}\in I$
Let $I$ a closed ideal of $C_0(X)$. Show that if $f\in I$, then $f^*=\overline{f}\in I$.
Tip: Let a sequence $(f_n)_{n=1}^\infty$ by $f_n=(f^*f)^{1/n}$, show that $f_n\in I$ and $\lim_{n}f_nf^*=f^*$.
...
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If the operator $U^2D_2+UD_1+D_1U^*+D_2U^{*2}$ is compact then $D_1$ and $D_2$ are compact
I need to show that if $U: \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ is the right shift operator and $D_1$ and $D_2$ are multiplication operators by the sequences $(a_n)_{n \in \mathbb{Z}}$ and $(b_n)...
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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 ...
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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 ...
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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).
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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General Spectral Mapping Theorem
It's well know from Functional Analysis that if $T$ is a operator in a $\mathbb{C}$-vector space and $p\in\mathbb{C}[z]$, then $\sigma(p(T))=p(\sigma(T))$.
It's true that for $p,q\in\mathbb{C}[z]$, $f=...
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Doubts related to joint-convexity
I have some doubts/clarifications related to jointly convex functions on positive semi-definite linear operators (denoted as $\mathcal{B}(\mathcal{H})_{+}$
Suppose we have a jointly-convex function $f:...
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Proving that if $A^*A + B^*B = 0$, then $A = B = 0$ for linear operators $A, B$
Is this proof valid?
$A^*A + B^*B = 0$ implies $A^*A = -B^*B$. But, for any linear operator $A$, we have $xA^*Ax \ge 0$. Since $xA^*Ax = - xB^*Bx$, $- xB^*Bx \ge 0$, so $xB^*Bx \le 0$. But also, $xB^*...
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Borel measurability of operator-valued map
Let $X$ be a separable real Hilbert space, $(\Omega,\mathcal{F})$ be a measurable space, and $x:\Omega\to X$ be a measurable map. Then for each $\omega\in \Omega$, we can define $T(\omega)=x(\omega)\...
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Trying to prove directly that the Laplacian is closed.
I am trying to prove directly that the Laplacian operator is closed on its maximal set. I have the following with some doubts.
Considering the operator $-\Delta$ defined on $D(-\Delta)=\left\{u\in L^2:...
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Convergence in the formulation of the spectral theorem
Let $\mathcal H$ be a complex (separable, if needed) Hilbert space and $B(\mathcal H)$ the ring of bounded operators on $\mathcal H$. I am interested in understanding the formulation of the spectral ...
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Tensor product and direct integral
Suppose that $A$ is a separably acting type $II_1$ von Neumann algebra with integral decomposition
$$A\cong \int_{X}^{\oplus} A_{x}\,d\mu(x),$$ where $A_x$ is a type $II_1$ factor a.e
Is it true that $...
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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 $\Delta:D(\Delta)\subset L^2\to L^2$ ...
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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 ...
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singular value decomposition of sum
Let $A,B$ be positive, linear trace class operators on some Hilbert space.
I would like to know if the following trace inequality for some $\mu>0$ is true
$$
\mathrm{Tr}\!\left(A\,(A+B+\mu I)^{-1}\...
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Let $T,S$ be unilateral shifts in $H,K$ and $A\in B(H,K)$ a contraction. If $S^*A=AT^*$, then why is $A$ a transposed infinite Toeplitz matrix?
Let $H, K$ be Hilbert spaces.
As the Toeplitz Matrix, I define an operator $P_n$ in the form:
$$P_n = \begin{pmatrix}
Q_0 & 0 & 0 & \ldots & 0 \\
Q_1 & Q_0 & 0 ...
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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 $K(x,y) = x^{-\frac{1}{2}} 1_{0 < y \leq x \leq 2y}(x,y)$, ...
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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....
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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 $T$ ...
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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 ...
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Hardy Spaces - Proving that the norm is well defined
I am self studying Banach Spaces of Analytic Function by Hoffman. In the Chapter 3 titled "Analytic and Harmonic Functions in the Unit Disc", the author defines the class $H^p$ for $1\le p \...
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Bound on spectral radii
Let $L$ be some linear and compact operator on some Hilbert space. And denote by $L^*$ the adjoint operator.
I need to bound for some $c>0$:
$$
\left\|(L^*L+c I)^{-\frac{1}{2}}L^*\right\|\leq1
$$
...
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About $\|1\|\neq1$ for operator algebras.
Let a normed algebra $A\ni1$ non-null. Suppose that $\|1\|\neq1$. Show that
(i) $\|1\|\geq1$;
(ii) $A$ have another $\|\cdot\|_n$ norm equivalent to $\|\cdot\|$ such that $\|1\|_n=1$ and $A$ still is ...
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General conditions for the invertibility of integral mappings
I am wondering when is an integral mapping invertible. More precisely, let $\mathcal{S}(\mathbb{R}^n)$ be the space of Schwartz smooth and rapidly decaying functions, its dual $\mathcal{S}'(\mathbb{R}^...
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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 ...
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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....
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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 $k \in C[0,1]^2$.
...
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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-...
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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 ...
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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 $\Delta\subset \sigma(x)$, ...
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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 ...
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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 ...
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1
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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 ...
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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 ...
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1
answer
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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
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1
answer
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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 ...
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Bounding $L^2$ operator norm of integral kernel
Say I have an integral operator $T$ with $g\in L^2$ such that
$$\int |Tg(\xi)|^2d\xi=\int g(x)\overline{g(y)}K_1(x,y)\,dxdy.$$
Is it correct that then, by repeated Cauchy-Schwarz, we have that
$$\| Tg\...
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1
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Hartree equation in density matrix formalism is equivalent to a system of Hartree equations
I want to prove that:
If the density matrix $$\gamma(t):=\sum_{j=1}^N|u_j(t)\rangle\langle u_j(t)|=\sum_{j=1}^N\langle u_j(t),\cdot\rangle u_j$$ solves $$i\partial_t\gamma(t)=[-\Delta+w*\rho,\gamma],$...
