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

1
vote
1answer
66 views

Fourier-Transformation of Operator

I have an operator $\hat{L}$ which gives $$\hat{L} f(x) = \lambda \cdot f(x)$$ where $\lambda$ is the eigenvalue. Now I Fourier-Transform my function $f(x)$: $$\mathcal{F}(f)(p) = g(p)$$ Question: ...
1
vote
1answer
79 views

Projecting self-adjoint operator onto closed subspace

Let $H$ be a complex Hilbert space and let $(Q, D(Q))$ be a closed, densely defined, positive semidefinite, Hermitian quadratic form on $H$. (That is, $D(Q)$ is a dense subspace of $H$, $Q$ maps ...
3
votes
1answer
84 views

Lifting a unitary to a partial isometry

What is an example of a unital $C^*$-algebra $A$ and an ideal $I$ such that some unitary element in $A/I$ cannot be lifted to a partial isometry in $A$? Or can it be shown using general properties of ...
4
votes
1answer
132 views

Integral with Dirac Delta

I've to compute this expression $$ \hat{H} = \frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2})\left[ \delta(\vec{r})\nabla_{\vec{r}}^2 ...
0
votes
1answer
37 views

Weyl Operators: Spectrum

Given a CCR-algebra $\mathcal{A}_{CCR}(\mathcal{H})$ over a Hilbert space $\mathcal{H}$. Then the Weyl operators are unitary: $$W(f)^*=W(-f)=W(f)^{-1}$$ Thus, their spectrum lies on the unit circle: ...
2
votes
1answer
122 views

Implying a positive definite operator

If we are given that $A:V \rightarrow V$ is an operator where $V$ is a real Hilbert space. If we are given that $A$ is bounded, strictly positive $\big(\langle Au,u \rangle > 0$ for all $u \neq ...
-1
votes
1answer
123 views

Bogoliubov Transformation

Let $\mathcal{A}_{CAR}(\mathcal{H})$ be a CAR algebra over a Hilbert space $\mathcal{H}$. Consider a linear $S$ and an antilinear $T$ both bounded operators acting on $\mathcal{H}$ satisfying: ...
1
vote
1answer
170 views

Spectral norm of a Hadamard product

Let $F$ be the $n\times n$ DFT matrix, i.e., $F_{i,j} = \exp\left(-\tfrac{2\pi(i-1)(j-1)}{n}\imath\right)$, for $1\le i,j\le n$ and $\imath =\sqrt{-1}$. Futhermore, let $\Vert \cdot\Vert$ and $\odot$ ...
1
vote
1answer
104 views

Tensor Product: Preliminary

Let $S$ and $T$ be bounded operators over a Hilbert space $\mathcal{H}$. Define their tensor product $S\otimes T$ as acting on $\mathcal{H}\otimes\mathcal{H}$ by $S\otimes T(x\otimes y):=Sx\otimes Ty$ ...
3
votes
2answers
436 views

Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentially selfadjoint if it contains a dense subset of analytic vectors?
1
vote
1answer
73 views

A question on tensor product of $C^{*}$ algebras

Let $A$ and $B$ be two $C^{*}$ algebras. Assume that every element of the minimal tensor product $A\otimes_{min} B$ is a finite linear combination of simple tensors $a\otimes b$. Can we say that ...
0
votes
1answer
321 views

Idempotent operators.

Apologies first. I am a physicist and my notations and rigour is probably lousy. If $P$ is an idempotent operator, $P^2 = P$, $P\neq \mathbb1$ and we have $\forall |\psi\rangle$ the relation, $P.L ...
1
vote
1answer
92 views

Application Closed Graph Theorem to Cauchy problem

Consider $E:=C^0([a,b])\times\mathbb{R}^n$ and $F:=C^n([a,b])$ equipped with the product norms. Consider $$ u^{(n)}+\sum_{i=0}^{n-1}a_i(t)u^{(i)}=f $$ with $$u(t_0)=w_1,\dots,u^{(n-1)}(t_0)=w_n \\ ...
1
vote
1answer
55 views

unbounded self-adjoint operator

Given an operator $T:D_1(T)\subset L^2 \rightarrow L^2$ and the same operator $T:D_2(T) \subset L^2 \rightarrow L^2$, such that the operator is both times self-adjoint and closed, with $D_1(T) \subset ...
1
vote
1answer
124 views

Mathieu differential equation

Given the operator $T (\psi)(x):= \psi''(x)-2q \cos(2x)\psi(x)$ with $T : D(T) \subset L^2[0,2\pi] $ I was wondering: What is the right domain $D(T)$ for this operator if we want to solve the ...
1
vote
2answers
261 views

Kadison's Inequality

Let $\mathcal{A}$ be a C*-algebra and $\omega$ a positive linear functional. Is there a simple proof for Kadison's inequality: $$|\omega(A)|^2\leq\|\omega\|\cdot\omega(A^*A)$$
0
votes
1answer
106 views

Momentum Operator: Selfadjoint Extensions

This might be a possible duplicate - please let me know if there is already a proof in another thread. Consider the momentum operator on $\mathcal{L}^2[0,2\pi]$: ...
4
votes
1answer
256 views

Proving the spectral theorem for unbounded self-adjoint operators

Let $A$ be (densely-defined) self-adjoint operator on a (complex) Hilbert space $H$. Then, the Cayley transform $U=(A-i)(A+i)^{-1}$ is a unitary operator, and $(A\pm i)^{-1} \in B(H)$. Using the ...
1
vote
1answer
57 views

Unbounded Operators: Notation?

For continuous a.k.a bounded operators we have $\mathcal{B}(X,Y)$ stressing on boundedness and $\mathcal{L}(X,Y)$ stressing on linearity entailing $\mathcal{C}(X,Y)$. Is there a notation for ...
1
vote
1answer
88 views

The spectral projection of a positive operator

Let $T_{n}\in B(H)$ be a positive operator on Hilbert space $H$ and $T_{n}\rightarrow 1_{H}$ in the strong operator topology. Now fix $\delta>0$ and let $P_{n}$ be the spectral projection of ...
5
votes
1answer
260 views

Positive Operators: Definition?

Definitions Given an operator algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ with $1\in\mathcal{A}$ Consider selfadjoint operators $A=A^*\in\mathcal{A}$. Define positive elements by: ...
2
votes
1answer
42 views

Approximations of compact operators

Let $(\xi_n)_{n=1}^\infty$ be a sequence in a Hilbert space $K$ convergent to some $\xi$. Suppose we have a compact operator $T$ on $K$ such that $T\xi = 0$. Can we find a sequence of compact ...
-1
votes
1answer
62 views

Number Operator closable on Fock Space?

In Bratelli Robinson the number operator in Fock space is defined as: $$\mathcal{D}(N):=\{\phi\in\mathcal{F}:\sum_{n=1}^\infty n|\|\phi_n\|<\infty\}\\ N:\mathcal{D}(N)\to ...
4
votes
2answers
94 views

Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
2
votes
2answers
296 views

Bounded, surjective linear operator between Banach spaces

How can I show that for a given surjective linear operator $T: X \to Y$ between Banach spaces, if there exists an $\epsilon > 0$ such that $||Tx|| \geq \epsilon||x||$ for all $x \in X$, then $T$ is ...
0
votes
0answers
35 views

A puzzling derivation about the expectation of [$\hat{X}$, $\hat{H}$]

a free particle of mass $m$, with Hamiltonian $\hat{H} = \frac {\hat{P}^2} {2m}$, where $\hat{P} = -i \hbar \frac{\partial} {\partial x}$. The commutative relation is given by $[\hat{X}, \hat{H}] ...
4
votes
1answer
57 views

Convergence of operator

I would like to know how to solve the following problem (since I didn't manage to solve it on today's exam): Let $A_h:L^1(a,b)\to L^1(a,b)$ be defined: $$A_h f(x)=\frac{1}{h}\int_x^{x+h} g(t) dt,$$ ...
0
votes
2answers
83 views

Spectrum of a bounded operator $T$ satisfying $T^n=I$

Let $\mathcal{H}$ be an infinite dimensional Hilbert space, suppose $T\in \mathcal{B}(\mathcal{H})$ is a bounded operator and suppose that $n$ is the smallest natural number so that $T^n=I$. Let ...
1
vote
0answers
59 views

Find the adjoint operator.

Consider the sequence space $\ell_p$ and S defined by $(1\leq p<\infty)$$$ S:\ell_p\to\ell_p:(x_1,x_2,x_3,\ldots)\mapsto(0,x_1,x_2,\ldots) $$ Find the $S^*$ operator.
2
votes
2answers
75 views

Computing the norm of operator when space is equipped with sup norm and $L^1$ norm

Let $\phi $ be the linear functional $\phi (f)=f(0)-\int^1_{-1}f(t)\:\mathrm{d}t$ a.Compute the norm of $\phi$ as a functional on Banach space $C[-1,1]$ with sup norm. b.Compute the of $\phi$ as a ...
4
votes
3answers
142 views

What is a predual of the Banach space of compact operators on $\ell^2$?

I am wondering if the space $K(\ell^2)$ of compact operators on $\ell^2$ can have a predual. Thank you in advance for your help.
1
vote
0answers
186 views

definition of unitary operator

Wiki says " A bounded linear operator $U: H \to H$ on a Hilbert space $H$ is called a unitary operator if it satisfies $U^{*}U=UU^{*}=I$ , where $U^{*}$ is the adjoint of $U$, $I$ is the identity ...
1
vote
1answer
96 views

Spectral radius of an operator equals its norm

Let $X$ be a Banach space and $A:X\to X$ a bounded operator. We know that the spectrum of $A$ is always included in the ball $B(0,|A|)$ and the spectral radious $r(A)$ is the smalest radius such that ...
0
votes
0answers
92 views

Spectrum of a bounded operator and Liouville's theorem

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator. We know that the spectrum of $A$ is not empty, because otherwise we find a contradiction by using the holomorphy of the function ...
1
vote
0answers
52 views

What's the name of this type operator?

If $H$, is a seperable Hilberspace, $E$ seperable Banach space. $(h_n)$ orthonormal basis of $H$. Let $T\in B(H,E)$. If we have the condition on $T$ that $$\sum_k \left\|Th_k\right\|^p <\infty,$$ ...
1
vote
1answer
79 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
2
votes
1answer
159 views

Riesz Lemma for reflexive spaces

I know the proof of Riesz Lemma: Let $Y$ be a closed (proper) subspace of a normed space $X$. Let $\varepsilon >0$. Then it exists an element $x \in X$ such that $||x||=1$ and $d(x, Y) \geq ...
0
votes
1answer
40 views

Give an example of a spanning set of $\ell^2(N)$ which is also a Bessel sequence but not a frame for $\ell^2(N)$

We know that in a finite dimensional Hilbert space, every spanning set is a frame, but this is not true for infinite dimensional space. It is easy to find an example which is a spanning set but not a ...
1
vote
1answer
152 views

Extension of a linear operator

Let $T$ be a linear operator defined on the space of the algebraic polinomials in $[0,1]$ (polinomials with rational coefficients) such that for each $k \in \mathbb{N}, T[x^k]=0$. Is it possibile to ...
1
vote
0answers
64 views

What is the definition of regular operator?

If $T$ is a bounded linear operator on a normed space $X$. What "$T$ is regular operator" means?
0
votes
0answers
41 views

Proving something is a convolution operator…

If we define the operator $K(a)=F^{−1}aF$ where $ F:L^2({\mathbb R})\to L^2({\mathbb R})$, is the fourier transform given by $$\left(Ff\right)\left(x\right)=\int_{{\mathbb ...
0
votes
0answers
26 views

Question about convergence of sum

Let $T\in B(H,E)$ where $H$ a seperable hilbertspace, $E$ a seperable Banach space. By parsevals identity $$\left\|T^*\right\|^2= \sup_{ \left\|x^*\right\|\leq 1}\left\|T^*x^*\right\|^2 = \sup_{ ...
1
vote
0answers
88 views

Skew adjoint operator with uncountable spectrum

Let $H$ be a Hilbert space. I just want an example of a skew adjoint operator $(A^*=-A)$ with uncountable spectrum. I also want an example for unbounded differential operators. The only example I ...
2
votes
1answer
85 views

$\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.

Suppose that $A$ is a bounded linear operator on a complex Banach space $X$ with resolvent set $\rho(A)$. If $C$ is a simple closed smooth curve in $\rho(A)$ such that $$ ...
5
votes
1answer
88 views

Show that $\|e^{tA}\| \le e^{t\|\Re (A)\|}$

Let $X$ be a complex Hilbert space, and let $A$ be a bounded linear operator on $X$. Define the real part of $A$ to be $\Re(A)=\frac{1}{2}(A^{\star}+A)$, and define ...
1
vote
1answer
53 views

Show that a set is compact.

Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for ...
0
votes
2answers
28 views

How to show something is a convolution operator?

I have the operator $W(a)$ defined by $$W(a)=F^{-1}aF$$ where $F$ denotes the fourier transform and $a$ is a function on $L^{\infty}$. I need to prove that this is convolution operator, but I don't ...
1
vote
2answers
40 views

What are non-tagential limits?

I'm reading this article where they use a set of functions, $H^{\infty}$, defined like this "Let $H^{\infty }$ be the closed subalgebra of $L^{\infty }({\mathbb R})$ that consists of all functions ...
2
votes
1answer
61 views

Relation between $A^{*}B=B^{*}A$ and $AB^{*}=BA^{*}$

Let $A$ and $B$ be two matrices. Can we say $A^{*}B=B^{*}A$ implies $AB^{*}=BA^{*}$? how about when $A$ or $B$ are normal? Any comments could be useful. Thanks.
0
votes
1answer
61 views

Sufficient condition for two operators being identical on Hilbert space

Considering two bounded linear operators $S,T$ in $\mathcal{B}(X)$, where $X$ is a complex Hilbert space. If $\def\norm#1#2{\langle {#1},{#2}\rangle} \norm{Sx}{x} = \norm{Tx}{x}$ for all $x\in X$, do ...