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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Calculate the matrix of a linear opertor that transforms a vector to a Hankel matrix

I would like to calculate the matrix associated to a linear operator $\mathbf{R}$ that transforms a vector $\mathbf{x}\in\mathbb{R}^N$ into a Hankel matrix $\mathbf{H}\in\mathbb{R}^{N-Q+1\times Q}$ ...
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Proving compactness of an operator $(Kf)(t)=\int_{0}^{\infty}k(t+s)f(s)ds$

I was trying to prove the compactness of the following operator: $K:L_2([0,\infty))\to L_2([0,\infty))$ $(Kf)(t) = \int_{0}^{\infty}k(t+s)f(s)ds$, given that the function $k$ is continous, and ...
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40 views

Extend and restriction of operator on $B(H)$

Let ‎$‎‎H$ ‎be a ‎Hilbert ‎space ,‎‎‎‎‎‎$‎‎B(H)$ ‎be ‎bounded ‎operators ‎on ‎‎$‎‎H$ ‎and ‎‎$‎‎K(H)$ ‎be ‎compact ‎operators ‎on ‎‎$‎‎H$‎. Assume ‎that ‎‎$‎‎M$ ‎is a ‎close‎d subspace of ‎$‎‎H$ ‎and ...
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3answers
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Example 9.18 from PMA Rudin

We know that $\gamma: (a,b)\to E\subset \mathbb{R}^n$ and $f:E\to \mathbb{R}^1$. Hence $f'(\gamma(t))\in L(\mathbb{R}^1, \mathbb{R}^1)$ since to any point $t\in(a,b)$ it corresponds some real ...
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97 views

K theory of finite dimenional Banach algebras

Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose $K_{0}$-group is a non trivial finite group?(I ...
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31 views

Comparison between Laplace, operator calculus and system of first order ODE

I am trying to understand those three methods to solve differential equations. I would like to know what actually are the differences between the three: Laplace calculus operator conversion to a ...
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1answer
52 views

Bounded Operators on a finite-dimensional Hilbert space - Linear combination of at most two unitaries and from a partial isometry to a unitary

Good day, In the lecture of functional analysis the proof of two statements were skipped as a task for us but I'm not sure how I approach these questions. a) Show that every partial isometry $V \in ...
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3answers
45 views

Derivative of linear transformation with confusing moment

After reading this part of Rudin's book i have one question: $A'(\mathbf{x})=A$ seems to me little bit weird because: 1) $A'(x)$ - it's derivative of operator $A$ at point $\mathbf{x}\in ...
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1answer
23 views

$L_1+L_2$ is close if $L_1\bot L_2$ are close sub-spaces of a Hilbert space $H$

$L_1+L_2$ is close if $L_1\bot L_2$ are sub-spaces of a Hilbert space $H$. While I do understand why it is true, I can't be completely sure how deduction is done here. I do know that if $\langle ...
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1answer
24 views

Definition of derivative for $n$D functions

After reading this text from PMA Rudin I have couple questions. 1) My first question about existence of 1-1 correspondence between $\mathbb{R}^1$ and $L(\mathbb{R}^1)$. Let's $\lambda \in ...
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1answer
99 views

Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$

Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$. What I need is a verification and guidance. I managed to show that the set is orthogonal. My ...
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1answer
34 views

Upper estimate of integral

I have the following integral defined on $\mathbb{C}$ of a function taking values in a Banach algebra. $$\int_{\mid z \mid =\mid \sigma(M) +\delta\mid }(z-M)^{-1}z^{n}d\bar{z}=M^n$$ where ...
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1answer
61 views

Spectrum of operator in $l_2$

Sorry guys, I have a problem with this exercise. Let T an operator in $l_2$ Hilbert space: $$ (\textrm{T}x)_1 = x_2 , $$ $$ (\textrm{T}x)_2 = 0 , $$ $$ (\textrm{T}x)_n = x_{n-1} - x_n ...
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22 views

An operator is linear and bounded on a hilbert space

an operator linear and bounded on a hilbert space Let H be the Hilbert space L^2(R), and assume that the continuous function g satisfies 0
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1answer
26 views

operator theory background

Mathematics is often divided into Analysis and Algebra. I want to know under which area Operator Theory lies. I have studied functional analysis where we studied operators on infinite dimensional ...
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1answer
58 views

If an operator have only Real eigenvalues + symmetric then it's self-adjoint?

I know that if an operator is self-adjoint then has Real eigenvalues but I'm not sure about the converse i.e. if it has only Real eigenvalues and is symmetric then the operator is selfadjoint. Is that ...
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1answer
26 views

Remark to theorem 9.8 from PMA Rudin

Let $\Omega$ be the set of all invertible linear operators on $\mathbb{R}^n$. Mapping $A\mapsto A^{-1}$ is obviously a $1-1$ mapping of $\Omega$ onto $\Omega$. This is excerpt from PMA Rudin. ...
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26 views

In a proof of the theorem about the abstract index group of a Banach algebra

The following is a proposition in the Banach Algebra Techniques in Operator Theory by Douglas: I don't quite understand the very last step of the proof. Let $\pi:G\to G/G_0$ be the cannonical ...
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30 views

*-isomorphism and spectrum

‎‎‎$A$ is a ‎‎‎‎$‎‎C^∗$-algebra and $P(A)$ is a set of projection of it. Assume that $A$ ‎admits a‎ ‎strictly ‎positive ‎element ‎‎‎‎‎$a$ ‎such ‎that ‎‎‎‎‎$‎‎‎‎σ(a)‎-\{‎0\}$ ‎is ‎discrete‎. I want to ...
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1answer
18 views

Is the distance of an element $a$ from a subspace $M$ always $||a-P_M a||$?

The distance of an element $a$ from a subspace $M$ is $||a-P_Ma||$? ($P_Ma$ is the orthogonal projection of $a$ on $M$). During the course of studying about Hilbert Spaces and The Operators Theory, I ...
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22 views

Some remarks about linear operators from PMA Rudin

Let $L(X,Y)$ be the set of all linear transformations of the vector space $X$ into the vector space $Y$. For $A\in L(\mathbb{R}^n,\mathbb{R}^m)$, define the norm $\lVert A\rVert=\sup\limits_{x: ...
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25 views

strictly positive elments $a$ when $‎‎‎\sigma(a) ‎\backslash‎ {0}‎$ ‎is ‎discrete

If ‎$‎‎A$ is a ‎‎$‎‎C^*$-algebra ‎and it ‎admits a‎ ‎strictly ‎positive ‎element ‎‎$‎‎a$ ‎such ‎that ‎‎$‎‎‎\sigma(a) ‎\backslash‎ {0}‎$ ‎is ‎discrete‎ then‎ Q1:‎$‎‎A$ admits ‎an ‎approximate ‎unit ...
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2answers
46 views

‎strictly ‎positive elements

Let ‎$‎‎A$ ‎be a ‎‎‎‎$‎‎C^*$-algebra‎. ‎$‎‎a\in A^+$ ‎is ‎strictly ‎positive in ‎$‎‎A$‎ ‎if ‎‎$‎‎‎\overline{aAa}=A‎$‎‎ *I know that if $A$ is unital, $a\in A^+$ is strictly positive iff $a\in ...
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1answer
23 views

minimal subnormal extensions of $u$

Q1:Does every $u\in B(H)$ admit a minimal subnormal operator?why? Q2:Two minimal subnormal extensions of $u$ are equivalent.
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32 views

subnormal operator

I know that ‎$‎‎u\in B(H)$ ‎is a‎ ‎normal ‎operator if ‎‎$‎‎uu^*=u^*u$‎. I ‎know ‎that ‎if ‎‎$‎u‎$‎‎ ‎is ‎subnormal ‎‎‎‎then ‎‎‎ ‎‎$‎‎uu^*‎\neq ‎u^*u$ ‎(like unilateral shift operator). ‎‎ My ...
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32 views

How to make sense of $(1-e^{tD})f$?

I'm sophomore student in college. Recently, I'm thinking about series expansion of operators. When I supposed that f is an $C^\infty$-function and D is the differential operator d/dt. According to ...
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1answer
25 views

Inverse bounded in a Banach space.

Let $X$ be a Banach space and let $A: X \rightarrow X $ be a bounded linear operator such that $A'(\tilde{X})=\tilde{X}$, show that $A$ has a bounded inverse (on its range). If someone could proof ...
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1answer
48 views

Spectrum $\sigma(T)$ of $T:l^1 \to l^1$ given by $T((a_j))=\left( \sum_{j=2}^{\infty} a_j \right) e_1 + \sum_{j=2}^{\infty} a_{j-1} e_j$

I'm considering the bounded linear operator $T$ on $l^1$ (the space of all absolutely convergent complex sequences) given by (with $e_k=(\delta_{kj})_{j=1,2,...}$) $$T((a_j))=\left( ...
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38 views

Nontrivial closed ideal of $\mathbb{B(H)}$, $\mathbb{H}$ is a non-separable Hilbert space.

$\mathbb{H}$ is a non-separable Hilbert space. Give an example of nontrivial closed ideal $I$of $\mathbb{B(H)}$, that is different from $\mathbb{B_0(H)}$ which is the ideal of compact operators. Any ...
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Does nonexpansive property in H-norm imply nonexpansive in 2-norm?

Suppose $\|f(x) - f(y)\|_H \le \|x - y\|_H$. In other words, $f$ is nonexpansive in the norm with respect to positive definite H: $\|z\|_H = z^T H z$. Can we then say something along these lines: ...
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19 views

Hilbert-Schmidt operator - converging norm series - Cylindrical brownian motion

I am reading about cylindrical brownian motion in the monograph of Prato and Zabczyk. For this construction a Hilbert-Schmidt operator is used, between to separable Hilbert spaces $U$ and $U_1.$ Let ...
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1answer
37 views

Proof of $\hat{\mathrm{O}}$ta's theorem

I'm trying to prove $\hat{\mathrm{O}}$ta's theorem : Let $A$ be a closed operator on a Hilbert space $H$ and $\overline{\mathcal{D}(A)}=H$. Suppose that $A\mathcal{D}(A)\subset \mathcal{D}(A)$ and ...
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1answer
28 views

Is T self adjoint and unitary?

Consider the Hilbert space $H=l^2 $over $\mathbb C$ .If $x\in l^2$,then $\displaystyle{ \sum_{i=1}^\infty}|x_k|^2<\infty$.If $x,y\in l^2$, the inner product is defined by $$\langle ...
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1answer
21 views

Does a symetric complex function $k(t,s)$ verify $\overline{k(t,s)}=k(t,s)$?

I am trying to figure out why an integral operator is self-adjoint. The operator is: $$K(f)=\int_{0}^{1} k(t,s)f(s)ds$$ From $L^2([0,1])$ to $L^2([0,1])$ and $0, \leq t,s \leq 1$ So I did a bit of ...
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15 views

Writing matrix representation of multiplication operator

For a given $m(x)\in L^2(0,1)$, let's write the multiplication operator $M\colon L^2(0,1)\longrightarrow L^2(0,1)$ as $Mf(x)=m(x)f(x)$. To write the matrix representation of this operator we need a ...
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19 views

Representing an operator in different bases

Say I have a random operator $\hat {A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ represented in the basis $\mathbf {e} = \left \{ \hat {e}_1, \hat {e}_2\right \}$ How should ...
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1answer
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Coherent states - operator algebra problem with physics motivation

Motivation: I have a mathematical problem motivated by quantum field theory in physics. It should be quite easy to prove, but for some reason I can't do it. Intro: Let there be operators $\hat{a_i}$ ...
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20 views

Change of basis for the matrix representation of an operator $L$

Suppose I have an operator, $L$, represented, in matrix form, in the orthonormal basis $\mathbf{e} = \left \{ \hat{e_1}, \hat{e_2} \right \}$, as $$L = \begin{pmatrix} 3 & \frac{3}{2} \\ ...
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1answer
32 views

Change of basis matrix for an operator

Suppose I have two operators, $D \equiv \frac{d}{dt}$ and $D^2 \equiv \frac{d^2}{dt^2}$, represented in matrix form by two different bases $\mathbf{e} = \left \{\cos (wt), \sin (wt) \right \}$ and ...
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1answer
53 views

Eigenvalue is giving me a null eigenvector, which contradicts its very definition

I'm computing the eigenvectors of an eigenvalue as part of the solution to this problem: $$(D^2 - I_2 \lambda_+) v_+ = 0$$ , where the eigenvalue is $\lambda_+ = w^2$ and $D^2$ is an operator given ...
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Spectrum of unitary elements of a Banach algebra

Unitary elements of a Banach space have been defined in this paper as follows: Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be ...
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50 views

About normal operators' spectral measure.

In fact, I want to prove a result in functional analysis about normal operators' spectral measure. It states as: If $N$ is a normal operator on Hilbert space $H$, then the spectral measure of $N$ is ...
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20 views

an isomorphism from $L^\infty(\mathbb{T})$ to $L^\infty([-1,1],\displaystyle\frac{2}{\pi } \sqrt{1-t^2}\mathrm{d}t)$

$\mathbb{T}$ is the boundary of unit ball.Consier $\phi:[-1,1]\rightarrow\mathbb{T},\phi(t)=exp^{2i(arcsint+t\sqrt{1-t^2})},t\in[-1,1]$. It is easy to check that $L^2(\mathbb{T})\ni f\mapsto f\phi\in ...
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1answer
31 views

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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1answer
40 views

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 ...
2
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1answer
54 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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1answer
50 views

$5$ 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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17 views

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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1answer
35 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 ...