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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Is a unital $*$-homomorphism preserving a state is one-to-one?

Let $M$ be a von Neumann algebra and let $\varphi$ be a faithful normal state on $M$. Suppose that $T \colon M \to M$ is a normal unital $*$-homomorphism preserving $\varphi$, i.e. $\varphi \circ T ...
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If every linear operator on $X$ is bounded then is $\dim X<\infty$?

I have started learning Functional Analysis: I have encountered a theorem which states that if If $X(NLS)$ is finite dimensional then every linear operator on $X$ is bounded. Is the converse ...
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59 views

the isomorphism of $L^\infty$ spaces

If we have an unitary operator from $L^2(\mathbb{T})$ to $L^2(X,d\mu)$ ,$\mathbb{T}$ is boundary of the unit ball ,$d\mu$is the Borel probability measure.is there an isomorphism between ...
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$5$ questions on the definition of the Gelfand triple

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb F\in\left\{\mathbb R,\mathbb C\right\}$, $\left\|\;\cdot\;\right\|$ be the norm induced by ...
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36 views

Operator and interpolation

I'm going to speak very informally, because i don't know how to formalize this stuff. If we have knots $X = \left\{ x_0, \ldots x_n \right\}$ and a given function $f$ belonging to an appropriate ...
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Show that the linear functional $\delta_{\lambda}(\varphi)=\varphi(\lambda),\delta: H^2(\Bbb{D})\to\Bbb{R}$ is continuous

Show that the linear functional $\delta_{\lambda}(\varphi)=\varphi(\lambda),\delta: H^2(\Bbb{D})\to\Bbb{R}$ is continuous where $\delta\in \Bbb{D}$, $\Bbb{D}$ is the unit disk, $H^2({\Bbb{D}})$ is a ...
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43 views

Do eigenvectors with pairwise distinct eigenvalues of a bounded, linear, nonnegative, symmetric operator on a Hilbert space build an orthogonal basis?

Let $H$ be a Hilbert space and $Q$ be a bounded, linear, nonnegative and symmetric operator on $H$ with finite trace. By the Hilbert–Schmidt theorem, there is an orthonormal basis ...
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44 views

Exponentiated Operators? ($e^{\hat{A}+\hat{B}} \ne e^{\hat{B}+\hat{A}}$)

Given, $$ e^{\hat{A}+\hat{B}} = e^{\hat{B}}e^{\hat{A}} $$ I then consider the series expansion of both exponentials. This then leads to a particular order of operation derived from the order of ...
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Is there a condition along with Strong convergence which gives convergence in norm?

Let $\{T_n\}$ be a sequence of operators in a Hilbert Space and $T_n$ converges to an operator $T$ in the strong operator topology. Is there any assumption which I can put on $\{T_n\}$ so that the SOT ...
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71 views

Proving the graph norm is indeed a norm

I was reading p.238-239 in the book enter link description here Example A.14 in that book motivates me to think of the following question. And have trouble verifying the following fact from ...
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Corollary of the Hahn-Banach theorem.

Recall the Hahn-Banach Theorem in Normed Vector Spaces: "Let $X$ be a NVS, and let $W$ be a linear subspace of $X$. Then, for any $f_W\in W'$ there exists an extension $f_X\in X'$ such that ...
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22 views

Trace class with bounded operator

If I assume to have a trace class $A$ and a bounded operator $B$, how can I show that: $tr AB = \sum \langle \psi_n|AB| \psi_n \rangle < \infty$ I do understand by sense, that this should be, as ...
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19 views

What is the definition of distinguished unit vector

What is the definition of distinguished unit vector? I guess it is the identity element,is it right?
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33 views

If there exists a bounded operator $T:X \to Y$ with $T^{-1}$ bounded then X is Banach iff Y is Banach

I have been asked to show that if there exists a bounded operator $T:X \to Y$ with $T^{-1}$ bounded then X is Banach iff Y is Banach. I have shown it for $T$ a linear operator. But I can't use the ...
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57 views

Show that there exists a certain operator in $L(H)$ where $H$ is a separable Hilbert Space.

Given a separable Hilbert Space $H$ and $\sum_{n=1}^{\infty} f_n$ an absolutely convergent series in $H$, I need to show that there is an operator $A \in L(H)$ such that $A(e_n)=f_n$, where ...
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24 views

Equality for functions in $H^2(\mathbb{R})$

I recently stumbled on the following equality: $$ \| (-\frac{d^2}{dx^2} + 1)^{1/2}g\|_{L^2} = \| g\|_{H^1}$$ for $g \in H^2(\mathbb{R})$. I tried to deduce the equality but failed (since I don't ...
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28 views

adjoint of operator?

let $H=L^2(0,1)$ (Hilbert space with usual scalar product )and the operator $A$ defined by : $D(A)=\{u\in C^1[0,1]:u(0)=\lambda u(1)\}$ where $\lambda\in\mathbb C$ and $Au=iu'$ my questions is : ...
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20 views

Operator norm and continuity

I've read in the solution of an exercise: "$T$ has a finite norm, thus $T$ is continuous". We are in a normed vector space $(V,||.||)$ and $T$ is a linear selfmap over the vector space $V$. The ...
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35 views

a proof about closable operator

I am self-studying the chapter of closed and closable operators. I have the following problem which I cannot find its proof. Let $A$ be a closable operator and denote by $B$ a closed extension. We ...
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Cyclic vector in $\ell^2(\mathbb{Z})$ space

Suppose we look at $\ell^2(\mathbb{Z})$, which contains vectors $c=(\dots,c_{-2},c_{-1},c_0,c_1,\dots)$ with $\sum|c(n)|^2<\infty$. Define the right-translation operator by $$R:\{c(n)\}\mapsto ...
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67 views

Prove: Expectation value is the weighted average

How do we mathematically prove that the expectation value of an operator is the weighted average, $$ \langle\hat{A}\rangle=\langle \psi|\hat{A}|\psi\rangle=\sum_{n}a_{n}P(a_{n}) \space \space \space? ...
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Isolated Eigenvalues on the Extensions.

I asked this question on Mathoverflow http://mathoverflow.net/questions/226484/isolated-eigenvalue-of-t-is-also-an-isolated-eigenvalue-of-overlinet and because of the comments apparently the answer ...
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42 views

Find the Left Inverse of an Operator

Suppose I have the following operator defined on the infinite line $L = \frac{\partial}{\partial t} + \lambda t$ Now, I can find the right inverse of this operator, $G(t,t_0)$, by solving the ...
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What is a biorthogonal system?

What does biorthogonal mean ? If they say let the system $l^1,l^2,l^3,...,l^n$ Be biorthogonal to the bases $x_1,x_2,x_3,...,x_n$ Of the kernel of $Λ$ So that $Λ$ Is a linear operator
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Finding the spectral decomposition of $\Delta= \frac{d^2}{dx^2}$ [closed]

What is the spectral decomposition of the operator $\Delta= \frac{d^2}{dx^2}$ in $(L^{2}(\mathbb R), dx)$? Thanks you in advance
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Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$

Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$, looking at $C_{(2)}[-1,1]$, with $L_2$ norm. I tried to look at a general polynomial $\sum_{i=0}^{98} ...
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37 views

positive elements and norm

If $A$ is a abelian $C^∗$-algebra and $a,b$ are elements in $A$ such that $0‎≤‎a‎≤‎1,0‎≤‎b‎≤‎1‎‎$ ‎‎ then $0‎≤‎\|a-b \|≤‎1‎$. My problem is:"Does the same hold if $A$ is not abelian?"
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Exponential Operators

What is wrong ? We have the following identity I try to check the equation, but I get a different answer I have considered the simplest case, in theory I should get two commutators but ... ...
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Exponential form for matrices

I'm trying to prove that for two commutative matrices $N$ and $M$, that $e^{N+M}=e^Ne^M$. I wrote using the binomial expansion and commutativity: ...
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Consequence of the polarization identity?

Here is a proof which I do not fully understand. Theorem : Let $H$ be a Hilbert space. A continuous linear map $T : H \rightarrow H$ is self-adjoint (hermitian) if and only if $$\big\langle T(x), ...
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51 views

What is the dual of the disc algebra viewed as a Banach space?

Let $A$ be the disc algebra, i.e., $A=\{f\in C(\bar{U}):f \text{ is holomorphic in }U\}$, where $U$ is the unit disc in the complex plane. The norm considered is the supremum norm. Are there any ...
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26 views

Validity of inequalities using integrals and absolute value

This question is similar to this one but the only response was pointing out mistakes in the solution. My goal is to determine whether the operator $T: C[0,1] \to C[0,1]$ defined by $Tx = ...
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Topelitz and matrix operators in $\ell_2$

Let $a,b$ satisfy $|a|,|b|<1$. We then define a vector $y = (\dots,b^2 ,b ,1 ,a ,a^2 ,\dots) \in \ell_ 1 (\mathbb{Z})$ with the ordering $y_0 = 1$. We define a matrix operator $Y$ by $$Yx = \sum ...
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Show that an operator is negative

I would show that, the operator $$A = \left(x_{4} \frac{\partial}{\partial x_{1} } -x_{1} \frac{\partial}{\partial x_{2} } \right) \frac{\partial}{\partial x_{3} } $$ is a negative operator on ...
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Relation kernels of linear operators

V is a vector space. T and U are two linear operators on V. For finite dimensional, dimker(TU)=dimker(T)+dimker(U) But, what about for infinite case?
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Property of inversion map on invertible operators

Given $X,Y$ two Banach spaces, I know the set of bounded operators $L(X,Y)$ is Banach with the operator norm $\|A\|=\sup_{\|x\|\leq1}\|Ax\|$. I know the set of bounded operators with bounded inverse ...
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33 views

A question on convergence.

If $u_n \rightarrow u$ in $L^p(\Omega)$ and suppose $u_n^{\frac{1}{p-1}}, u^{\frac{1}{p-1}} \in L^p(\Omega) \forall n$ then can it be said that $u_n^{1/p-1}\rightarrow u^{1/p-1}$ in $L^p(\Omega)$?.
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Do there exist bounded operators with unbounded inverses?

I have just been introduced to the concept of invertibility for bounded linear operators. Specifically, we defined a bounded operator $A$ to be invertible if there exists a bounded $A^{-1}$ which is ...
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Modified shift operator is compact.

For the operator $$T(\eta_j) = \frac{\eta_{j+1}}{j}$$ on Hilbert Space $H$ where $(\eta_j)$ is a basis. Show it is compact. Can this work? Define $$f = (\eta_j)_{j \geq 1}$$ $$T_N(f) = ...
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Spectrum and resolvent of an operator

So for the operator $A:l_2(\Bbb C)\to l_2(\Bbb C)$ defined as: $$A(x_1,x_2,\cdots,x_m,x_{m+1},x_{m+2},\cdots) = (x_1,x_2,\cdots,x_m,0,0,\cdots)$$ We can find the adjoint operator $A^*$ by looking ...
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Weak convergence = norm convergence for trace class operators?

Given a (separable) Hilbertspace $H$, I look at the traceclass operators $\mathfrak{S}_1$. I recall the fact that the weak convergence implies norm convergence in the sequence space $\mathcal{l}^1$. ...
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Show that the following operator (on a Hilbert space) is continuous.

"Let $\mathcal H$ be a complex Hilbert space and let $y\in\mathcal H.$ Show that the linear transformation $f:\mathcal H\to\mathbb C$ defined by, $f(x)=\langle x,y\rangle$ is continuous." ...
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Approximation property for Banach space and $l^{p}$

Let's consider a compact operator $T: X \rightarrow l^{p}, 1 \leq p < \infty$. I would like to check, whether it's possible to approximate $T$ by the operators of a finite rank with respect to an ...
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Convergence in $L^p$ of product space implies convergence in each space?

Reading a paper by EM Stein (On limits of sequences of Operators, Ann of Math, 1961), the author proves that a certain sequence of functions $F_n(x, t)$, where $x$ belongs to a probability space $(X, ...
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What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds?

Consider the operator $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ acting on a function space $\mathbf{F}(M)$, given by the set of functions $\phi:M\to\mathbb{R}$ whose values go to zero at infinity (at the ...
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An operator which moves on the boundary

Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis in $H$. Let $E_0$ be a countable subset of $E$ and $p$ be the projection onto the space generated by $E_0$. Let ...
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Show that the following operator is not a surjection.

"Take $X=C([0,1])$ with the uniform norm, $\|f\|=\sup_{x\in[0,1]}|f(x)|$, and define the operator $T:X\to X$ by, $$T(f)(x)=f(x)-\int_0^1f(s)ds$$ Show that $T$ is not a surjection". Here is ...
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97 views

Find the norm of the following operator.

Take $X=C([0,1])$ with the uniform norm, $\|f\|=\sup_{x\in[0,1]}|f(x)|$, and define the operator $T:X\to X$ by, $$T(f)(x)=f(x)-\int_0^1f(s)ds$$ Find $\|T\|$. I was hoping to solve this ...
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40 views

A system of equations

Let $H$ be a non-separable Hilbert space. Assume $E$ is an orthonormal basis in $H$. Let $E_0=\{e_n\}$ be a countable subset of $E$ and let $\{\zeta_n\}$ be a bounded sequence in $H$. Let $E_1$ be a ...
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Norm of the inverse of a map $\ell^2\to\ell^2$

Let $Au_i=u_{i+1}-(2-\beta)u_i+u_{i-1}$ whith $u\in \ell^2=\{(u_i)_{i\in \mathbb Z}, u_i\in \mathbb R:\sum_{i\in \mathbb Z}u^2_i<+\infty\}; \beta>0$. How to compute $||A^{-1}||$ or estimate it? ...