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.

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If $A$ is the Laplacian on $H^2(0,1)∩H_0^1(D)$, then the fractional power space $\mathfrak D(A^{r/2})=H_0^r(D)$ for all $r\in\mathbb R$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
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If $G$ is the Green's function of the Laplacian $A$ and $L$ is the integral operator with kernel $G$, then $L$ is the inverse of $A$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
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25 views

How do we compute the Green's function of the Laplacian?

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
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49 views

What is the proof that linear operators can be treated as variables?

I understand what a linear operator is, but I don't understand why you can just treat it as a variable.
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34 views

Bilinear Form in Definition of Adjoint Operator

I'm working through some material in Analysis Now by Pedersen, and I'm a little confused about how the adjoint to an operator is defined. Pederson defines it by: If $X$ and $Y$ are normed spaces ...
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42 views

Example of a Projection Operator in $\mathbb{R^3}$

I'm looking for an operator $\hat P$ in $\mathbb{R^3}$ such that $\hat P^2=\hat P$ that is also Hermitian
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29 views

Is a bounded, linear, nonnegative and symmetric operator with finite trace on a Hilbert space Hilbert-Schmidt?

Let $U=(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $Q$ be a bounded, linear, ...
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1answer
57 views

Approximate unit for a certain C*-algebra

Let $A$ be a C*-algebra and $p$ a projection in $A^{**}$. To prove $p$ is the smallest unit for $B: = \{a\in A; pap=a\}$, suppose $\{u_i\}$ is an approximate unit for $B$. It's easy to see $q: = w^*-\...
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50 views

Trace of multiplication operator on $L^2(\mathbb{T})$

Let $H=L^2(\mathbb{T})$, where $\mathbb{T}$ is the Torus. Consider a multiplication operator with a sufficiently nice function $f$. Is there somehow a formula like $$\mathrm{tr} M_f = C \int_{\mathbb{...
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24 views

Prove Linear Operator $(T_{w}f)(x) = |\dot{w}|^{\frac{1}{2}} f(w(x)) $ Is Unitary

Let the inner product be given by: $$\langle f,g \rangle = \int_{-\infty}^{\infty} f(x)g^{\ast}(x)dx$$ Then I want to show: $\langle T_{w}f, T_{w}g \rangle = \langle f,g \rangle$ So we have: $$\...
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32 views

what is the operator name that for positive value returns the same and for negatvie returns zero?

there is an operator that currently I see in mechanical engineering topics that for positive values returns the same quantity inside, and for negative values, returns zero , it's like <> but a bit ...
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35 views

Operator theory problem

Show that operator $ T : C([0,1]) \to C([0,1]) $, where $$ Ty(t) = \int_{0}^{1} |x-t|^{-1/2}y(x) \ dx. $$ Calculate the norm T. Thank for help.
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Show that $S_n \to S $(weakly) and $T_n \to T$ strongly implies $S_nT_n \to ST$ weakly

Let $X,Y,Z$ be Banach Spaces. Let $T_n,T \subset BL(X,Y), S_n,S \in BL(Y,Z)$. Show that a) $S_n \to S $(weakly) and $T_n \to T$ (strongly) implies $S_nT_n \to ST$ (weakly) b) $S_n \to S $(uniformly)...
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Is the Hankel matrix defined by $\ln(k)/k$ a bounded operator on $\ell^2$?

I call a sequence $(b_k)$ of positive real numbers a Hilbert kernel if there is a constant $C > 0$ such that $$ \sum_{k=1}^\infty\sum_{j=1}^\infty b_{k+j}c_kc_j\,\le\,C\sum_{k=1}^\infty c_k^2 $$ ...
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2answers
48 views

Functional calculus for unitization of an algebra?

I have been stuck on this problem for a week now: "Let $A$ be a Banach algebra without identity, let $a\in A$, and let $f$ be holomorphic on a neighborhood of $\sigma(a)$, so that $f(a)\in A^\# $ is ...
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32 views

Functional analysis, linear operator problem

a) Let $D$ be closed subspace of $l^1$ defined as follows $$ D : = \left\{ x=(s_n)_{n \in \mathbb{N}} \in l^1 : \sum_{n=1}^\infty \left\vert {s_n \over n} \right\vert^2 < \infty \right\} $$ Is ...
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28 views

Proving an identity of the Lebesgue integral of a sublinear operator

Let $T$ be a sublinear operator. Suppose that $f\in L^{1}(\mathbb{R}^{n})$ and $A$ is a set of finite Lebesgue measure. Then I want to prove that for all $0<p<1$, we have $$\int_{A}|T(f)(x)|^{p}...
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67 views

The strong topology on $U(\mathcal H)$ is metrisable

The strong operator topology on a Banach space $X$ is usually defined via semi-norms: For any $x \in X$, $|\cdot|_x: B(X) \to \mathbb R, A \mapsto \|A(x)\|$ is a semi-norm, the strong topology is the ...
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65 views

The reduction of nilpotency order of nilpotent elements of $C^{*}$ algebras

Assume that $A$ is a unital $C^{*}$-algebra. Let $a\in A$ be a nilpotent element with $$a^{k}=0,\;\;k>1.$$ Are there two elements $x,y\in A$ with $a=xy,\;\;(yx)^{k-1}=0$? Motivation for ...
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Is the range of a self-adjoint operator stable by its exponential?

Let $H$ be an Hilbert space, and $A \in L(H)$ be a bounded linear self-adjoint operator on $A$. We assume that $R(A)$, the range of $A$, is not closed. Is it true or not that $R(A)$ is stable by $e^{-...
2
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57 views

If $Q$ is an operator on a Hilbert space with $Qe_n=λ_ne_n$ for all $n$, then $Q^{-\frac 12}e_n=\frac 1{\sqrt{λ_n}}e_n$ for all $n$ with $λ_n>0$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $\mathfrak L(U)$ be the set of bounded and linear operators on $U$ $Q\in\mathfrak L(U)$ be nonnegative and symmetric $(e_n)_{...
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51 views

If $Q$ is an operator on a Hilbert space $U$, $(e_n)$ is an ONB of $U$ consisting of eigenvectors of $Q$, then $(Q^{1/2}e_n)$ is an ONB of $Q^{1/2}U$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ ...
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24 views

Solvable Schrödinger operator

I am currently looking for an example of a 1-dimensional Schrödinger Operator $-\frac{d^2}{dx^2} + V(x)$ with the following properties: 1) V(x) must be integrable 2) $-\frac{d^2}{dx^2} +V(x)$ must ...
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On connection of distance to finite rank operators and singular values.

Im trying to understand why the following; $s_n(T) = \inf\big\{\, \|T-L\| : L\text{ is an operator of finite rank }<n \,\big\}$ where $s_{n}$ are nth singular values, is a plausible claim to ...
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38 views

What are eigenvalues/eigenfunctions of a “pointwise product” operator

Let us consider the Hilbert space $l^2([0,1])$ with inner product $<u,v>=\int_0^1 u(x)v(x)\mathrm dx$. We define a pointwise product operator $A$ as $(A\circ u)(x)=a(x)\cdot u(x)$, where "$\cdot$...
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28 views

Is there a sample-path continuous stochastic process whose sample paths do not almost surely lie in an RKHS?

Let $f$ be a mean zero second-order stochastic process with continuous covariance function $k$, that is indexed on a separable metric space $\mathcal{X}$ and that is sample-path continuous. Can we ...
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Can $xy$ and $yx$ lie in different connected components of the group of invertible elements of an algebra?

What is an example of a Banach or $C^{*}$ algebra $A$ which has two invertible elements $x, y$ such that $xy$ can not be connected to $yx$ in $G(A)$, the space of invertible elements of $A$. A ...
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21 views

Image density in spectral theory

The operator $T$ is $\dfrac{d}{dt}$ and $$\left\{\begin{array}{lc}x'(t)-\lambda x(t)=-y(t)\\ x(0)=0\end{array}\right.$$ and the domain of $T$ is $D(T)=\{x\in L^2(0,\infty):\; x\; \text{absolutely ...
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Given a linear Hilbert-Schmidt embedding $ι$ between Hilbert spaces, prove that $ιι^*$ is a bounded, linear operator with finite trace

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $U_0:=Q^{\frac 12}(U)$, $$\langle u,v\rangle_0:=\langle Q^...
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1answer
39 views

Showing that a linear operator is closed

Consider the linear operator $A:D(A)\subset X\to X$. I want to show that $(\lambda I-A)$ is closed given that $(\lambda I-A)$ is invertible. We know that $(\lambda I-A)^{-1}$ is closed. Now if we let ...
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62 views

$T\colon L^2[0,1] \to L^2[0,1]$ be a bounded linear map of Hilbert spaces such that if $f\in L^2[0,1]$ is continuous then so is $Tf$. [closed]

Let $T\colon L^2[0,1] \to L^2[0,1]$ be a bounded linear map of Hilbert spaces such that if $f\in L^2[0,1]$ is continuous then so is $Tf$. Show that there is a constant $C$ such that $$\sup_{x\in[0,1]}|...
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If $Q$ is an operator on a Hilbert space $U$, $U_0:=Q^{1/2}(U)$ and $(e_n)_n$ is a basis of $U_0$, then $u↦\sum_na_n(u,e_n)_0e_n$ is an embedding

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $Q$ be a bounded, linear, ...
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35 views

How to show the lifting property of an operator described in its ellipticity

Consider Soblev spaces as special cases of Besov spaces characterized by wavelets, $H^s = B^s_{2,2}$. I want to prove the following statement. For a positive definite self-adjoint operator $A$, if ...
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Are all operators to or from $\ell_1$ completely continuous?

Let $E$ and $F$ be two Banach spaces, and let $T \in \mathcal{L}(E, F)$. Consider the following property (P). For every weakly convergent sequence $(u_n)$ in $E$, $u_n \rightharpoonup u$, then $...
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1answer
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Can all compact subsets of $\mathbb{C}$ be spectra of a bounded operator on $C[0,1]$?

Let $K$ be a non-empty compact subset of $\mathbb{C}$, the complex field. Does there exist an operator $A\in \mathscr{B}(C[0,1])$ such that $\sigma(A)=K$? $A$ is a multiplication operator iff $K$ is ...
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15 views

Existence of nuclear dominating positive definite kernel

Let $\mathcal{X}$ be a metric space and $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ a continuous positive definite kernel. Can we always find a positive definite kernel $r$ such that $r \gg k$ (...
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2answers
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Is a self-adjoint operator continuous on its domain?

Let $H$ be a Hilbert space, and $A : D(A) \subset H \rightarrow H$ be an unbounded linear operator, with a domain $D(A)$ being dense in H. We assume that $A$ is self-adjoint, that is $A^*=A$. Since $...
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Binormal operator - equivalent definitions?

I have seen two different definitions of a binormal operator A. A is unitarily equivalent to a block 2x2 matrix of commuting normal matrices. AA* commutes with A*A. I am hoping these definitions ...
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117 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, ...
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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 ...
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150 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 ...
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$V(A)$ 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 ...
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64 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 Hilbert space $\mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}$. $\...
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1answer
78 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 ...
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1answer
57 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 \, ...
2
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1answer
67 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. ...
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24 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 $S$...
3
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2answers
80 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 ...
2
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1answer
41 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 $\sup_{||x||_{\ell^2}=1}\sum_{l=1}^\infty||T_l(x)||^2<\...
4
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1answer
65 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 $e^{*}$?