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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Example of operator with spectrum equal to $\mathbb{C}$?

In my Functional Analysis course, we proved that for a (possibly unbounded) operator $T$ that is densely defined, closed, and symmetric, exactly one of the following four occurs: $\sigma(T) = \...
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21 views

How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$?

Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha \log(A)...
3
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2answers
105 views

Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$ Some definitions and results of the ...
119
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5answers
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How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
3
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1answer
48 views

‎Jointly ‎continuous of product in $B(H)$

‎Let ‎$‎B(H)‎$ ‎be the set of‎ ‎bounded ‎operators ‎on a Hilbert space ‎‎$‎‎H$.‎ ‎ I ‎know that ‎$u_{‎\alpha‎}‎\longrightarrow u‎$‎‎‎ ‎in ‎S.O.T ‎if ‎and ‎only ‎if‎ ‎$u_{‎\alpha‎}(x)‎\longrightarrow ...
1
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1answer
26 views

Largest invariant subspace

If $A$ is a $n\times n$ matrix with complex entries, denote by $inv(A)$ the dimension of a largest dimensional non-trivial invariant subspace of $A$. What is: $$\inf\{inv(A): A\in\mathbb{M}_n(\mathbb{...
2
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1answer
28 views

Small perturbations and eigenvalues

Suppose $A$ is a $n\times n$ matrix. Given $\epsilon>0$, can one find a rank one matrix $B$ with euclidean norm at most $\epsilon$ such that $A+B$ has $n$ distinct complex eigenvalues? Given a non-...
0
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1answer
57 views

Double dual of the space of bounded operators on Hilbert space [duplicate]

Every Banach space $X$ is canonically, isometrically embedded in its bidual $X^{**}$. But it is not always $1$-complemented in the bidual: for example, there is no projection from $\ell_\infty$ onto $...
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0answers
53 views

Eigenfunctions of integral operator

I am faced with the problem of calculating the eigenfunctions for an operator of the form: $(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy $ Does anyone know for which functions (or types of ...
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0answers
51 views

There are infinitely many projections

Can anyone explain please why such projections are infinitely many?
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2answers
57 views

Prove that there is no norm for to make this mapping continuous

I am dealing with an exercise which is as follows: Show that there is no norm such that the set of all the mappings $T_a$ which map every element $f\in C(\mathbb{R}, \mathbb{R})$ (where the latter is ...
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0answers
19 views

Brief moment from Definition 9.30 from Rudin's PMA

I have some problems with claim which is marked with red line. Claim: Let $A\in L(\mathbb{R}^n)$. Then $A$ is invertible if and only if $\text{rk}A=n$. Proof: $\Rightarrow$ We know that $A\in L(\...
2
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1answer
53 views

Square Root of the shift operator indexed by $\mathbb{Z}$

My question is very similar to this question, but instead of indexing by $\mathbb N$ I am indexing by $\mathbb Z$. Consider the shift operator $T : \ell^1(\mathbb Z) \to \ell^1(\mathbb Z) $ given by $...
2
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0answers
39 views

Conditions under which an Convolution operator is normal.

I have a possibly complex valued convolution operator given by $ \int_{\mathbb{R}}K(x-y)f(y)dy$ I know that the operator is self-adjoint if $K(x)=\overline{K(-x)}$ holds. But are there softer ...
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3answers
109 views

Examples of bounded linear operators with range not closed

I've been trying to get some intuition on what it means for a bounded linear operator to have closed range. Can anyone give some simple examples of such an operator that does not have closed range? ...
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1answer
30 views

Weak operator topology convergence of hermitian operators

Let $\{A_i\}$ be a net of hermitian operators on a separable Hibert space $\mathbb{H}$ and suppose that there is a hermitian operator T such that $A_{i}\le T$ for all i. If $\{<A_i h,h>\}$ is an ...
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2answers
42 views

Prove equality of norms of operators

Let $e_i$=${\{\delta_{k,i}}\}_{k\ge1}$ $\in$ $l_2$, $i\ge1$, $A_n$ and $B_n$ - operators that are defined like this: $A_n\{x_i\}_{i\ge1}$ = $x_ne_1$, $B_n\{x_i\}_{i\ge1}$ = $x_1e_n$ $(\{x_i\}_{i\...
3
votes
1answer
35 views

property of the Gelfand transform: why does an isometry map closed sets to closed sets?

The following is a theorem about self-adjoint subalgebra of $C(X)$ where $X$ is compact Hausdorff and the first half of its proof: Here are my questions: Why $\Gamma:\mathfrak{U}\to C(M_{\...
3
votes
1answer
129 views

Closure of an Operator in $l^2$

Let $l^2$ denote the Hilbert space of all complex sequences $\phi = (\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j|^2 < \infty$. Consider the linear subspace of $l^2$ defined by \...
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1answer
33 views

continuity of a function and net convergence

The following is a statement and its proof in the Banach Algebra Techniques for Operator Theory by Douglas: I don't understand the last part of the proof. In order to show that $f$ is continuous, ...
0
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1answer
30 views

Are linear and continuous mappings between locally convex vector spaces bounded?

I know that continuity and boundedness of linear mappings between normed vector spaces are equivalent, but does the same hold true for locally convex vector spaces? If so, how can we prove it?
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2answers
64 views

Adjoint of an Operator in $l^2$

Let $l^2$ be the Hilbert space of all complex sequences $\phi =(\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j |^2 < \infty$. Set $D= \{ \phi \in l^2 : \sum_{j=0}^{\infty} j |\...
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0answers
13 views

Closable Multiplication Operator

I have the operator $M:Dom (M)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$, $Mf(x)=m(x)f(x)$, where $m$ is a continuous function and $Dom(M)=\{f\in L^2(\mathbb{R}^N)| mf\in L^2(\mathbb{R}^N)...
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39 views

C*-algebras: Proofs on $C_0(X)$

I'm looking to prove the following but am stuck, please can you help me? $C_0(X)$ is isomorphic as a C*-algebra to $C_0(Y)$ if and only if X is homeomorphic to Y, where X and Y are locally compact ...
0
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1answer
42 views

Regarding integral operators being contractions

I have two half-questions that tie into one another. Suppose $T$ is an operator on $C([0, 1])$ defined by $$(Tu)(t) = \int_0^t (u(x))^2dx.$$ Show that T is not a contraction on the closed unit ball ...
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1answer
38 views

Open sets in the unitary group $ U(\mathcal{H}) $ of a Hilbert space $ \mathcal{H} $.

Let $H$ be an infinite dimensional Hilbert space and let $(x_i)_1^\infty$ be an orthonormal basis for $H$. Consider $U(H)$ the unitary group of the continuous unitary operators on $H$. Equip $U(H)$ ...
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1answer
33 views

Find Riesz representation of $\phi=f({1\over 2})$

"Let $\rho$ be a space of complex polynomial and define $<f,g>={1\over 2\pi}\int_{0}^{2\pi}f(e^{it})\overline{g(e^{it})}dt$ for $f,g:\rho\to \Bbb{C}$. Let $\phi$ be a linear functional on $\rho$...
0
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1answer
50 views

Operator Norm: Why is this formulation so rarely seen?

There are many equivalent definitions for the norm $||T||$ of a linear operator $T:X\to Y$. Citing the Wikipedia and some books about functional analysis, the equivalence of definitions are often ...
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8 views

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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0answers
57 views

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 $\int_{...
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0answers
42 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
48 views

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 number. ...
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1answer
117 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 ...
3
votes
1answer
87 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
58 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 $\mathbf{x}\in \mathbb{R^n}$...
2
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1answer
25 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 z_n-...
0
votes
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 \mathbb{R}...
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1answer
142 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
35 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 $d\bar{z}=\...
3
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1answer
69 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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1answer
45 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
38 views

Sequences of operators in $B(H)$ which converge in the weak operator topology

Let $H$ be a Hilbert space. Suppose that $\{u_n\} \subseteq B(H)$ is a sequence which converges to some operator $u$ in the weak operator topology, which means that for all $x,y\in H$ one has ‎$$‎‎‎‎\...
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1answer
68 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 ...
0
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1answer
35 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. But ...
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35 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 ...
0
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1answer
34 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 ...
0
votes
1answer
24 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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0answers
25 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: |x|\...
0
votes
0answers
27 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 ‎‎$...
2
votes
2answers
52 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 $a\in Inv(A)$...