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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functional calculus on a set of normal elements is continuous

Let $K$ be a compact subset of $\Bbb C$. Let $A_K$ denote the set of all normal elements $x$ with $\sigma_A(x)\subset K$. If $f$ is a continuous function on $K$, then the functional calculus :$x\in ...
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34 views

Confusion with Riesz

I was reading this article here and it claims that the covariance operator from a Hilbert space to itself exists by the "Riesz representation theorem". I don't seem to see the link between the Riesz ...
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23 views

Find an operator $Z$ in $H^1(0, \infty)$ with $\langle u,Zv\rangle = \int \bar{u}v dx$

I'm working with operators associated to bilinear forms. What I need to find is a continous, linear operator $T$ defined on $H^1((0, \infty))$ [note that $H^1 = W^{1,2}$ is the Sobolev space] such ...
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35 views

polar decomposition of multiplicative operator on L^2 induced by identity function.

We know that every operator in B(H) has a polar decomposition. $T=VP$ that $P=|T|$ and V is a partial isometry with initial space closure of ImP and final space ImT. How can i obtain polar ...
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1answer
15 views

Calculating the form domain of an operator

I am reading the book "Mathematical Methods in Quantum Mechanics" by Gerald Teschl and just came across the concept of a form domain. It is defined for non-negative operators i.e $<\phi,A \phi> ...
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32 views

Commutant of a C*-subalgebra of B(H)

In operator theory, we can prove that the commutant of $B(H)$ is $\mathbb{C} I$, where $I$ is the identity map. But a book states that every $C*$-subalgebra of $B(H)$ that contains the compact ...
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51 views

Does $\|f(T^*T)T^*\|_\infty = \|f(T^*T)(T^*T)^{\frac{1}{2}}\|_\infty$?

If $T:X\to Y$ is a compact operator and $X,Y$ are some Hilbert spaces, can we say that $\|f(T^*T)T^*\|_\infty = \|f(T^*T)(T^*T)^{\frac{1}{2}}\|_\infty$, where $T^*$ is its adjoint and $f$ some ...
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36 views

Find the operator norm

Let $T : \ell^2 \to \ell^2$ (involving complex numbers) be defined by $$ Tx := (x_1, x_1+x_2, x_3, x_3+x_4, x_5, x_5+x_6, \ldots). $$ What is $\|T\|$? Essentially I've tried : To find $M \geq 0$ ...
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26 views

Derivative of a function which is treated as a variable

I have got a function $f=f(x)$. The derivative is $\partial_xf$. There are applications in which it is reasonable to treat $f$ as another variable in a larger context. In my application I now need an ...
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1answer
32 views

an exercise about the projections.

There is an exercise in operator theory that says: If P and Q are projections on H that $||P-Q||<1$ then dimension of ImP and ImQ are the same. i cant understand what is the relation between the ...
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63 views

Norm of the Linear (Integral) Operator on $C[0,1]$

There may be some similar questions being asked but I didn't see anything that was quite what I was looking for so I'll ask the question here. Let $X = (C[0,1],||.||_{\max})$ and $T:X \to X = ...
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1answer
22 views

Homeomorphism between locally compact space $\Omega$ and maximal ideals space of $C_0(\Omega)$

the following is a proposition: If $\Omega$ is locally compact and $\Sigma$ is the maximal ideal space of $C_0(\Omega)$, then the map $x\to \delta_x$ is a homeomorphism. To prove it, the author ...
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2answers
79 views

Is there any multiplicative linear functional on B(H)?

If A is a Banach algebra, we say that $\Phi: A \longrightarrow \mathbb{C}$ is a multiplicative linear functional if $\Phi$ is nontrivial, linear and $\Phi(xy)=\Phi(x)\Phi(y)$. It is easy to see that ...
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36 views

A Representation of $C(X)$ is a positive map.

I quote this excerpt from Conway: "A representation $\rho:C(X) \rightarrow \mathcal{B(\mathcal{H}})$ is a $\ast$-homomorphism with $\rho(1)=1$. Also, $\|\rho\|=1$. If $f\in C(X)_+$, then $f=g^2$ ...
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73 views

Closed unit ball of B(H) is not compact in strong operator topology of B(H).

In operator theory we prove that closed unit ball of B(H) is compact in weak operator topology and is closed in strong operator topology. But a book of operator theory states that closed unit ball of ...
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49 views

The map $T\longmapsto \|T\|$ is not continuous in the strong operator topology of $\mathscr B(H)$

In the context of Strong and Weak operator topologies on $\mathscr B(H)$ there is an statement that says: the map on $\mathscr B(H)$ that $T\longmapsto \|T\|$ is not continuous in the strong operator ...
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1answer
39 views

an exercise about the spectrum of an element in Banach algebra.

An exercise of Banach algebra, section of spectrum has wanted the proof of this statement: Let $A$ be a Banach algebra and $x\in A$. Show that for every open set $U$ in $\mathbb{C}$ that contains ...
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47 views

Spectral radius of a normal element in a Banach algebra

I know that if $A$ is a C*-algebra, then $||x||=||x||_{sp}$ for every normal element $x\in A$. I like to find an example of a normal element $x$ in a involutive Banach algebra with $||x||\neq ...
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51 views

Is a contractive algebraic homomorphism between unital $ C^{*} $-algebras a unital $ C^{*} $-algebraic homomorphism?

We know that a $ C^{*} $-algebraic homomorphism from a unital $ C^{*} $-algebra $ A $ to a unital $ C^{*} $-algebra $ B $ is a linear multiplicative mapping that preserves units and respects the $ * ...
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1answer
27 views

Why is this statement true for two equivalent projections in $B(H)$?

In a book of operator theory it is stated that two projections $P$ and $Q$ in a von Neumann algebra $A$ are equivalent if there exist $V$ in $A$ that $V^*V=P$ and $VV^*=Q$. After this definition, it ...
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78 views

Spectrum of an element of a non unital C*-algebra

I know that spectrum of an element $x$ of a unital C*-algebra is nonempty. I like to find an example of a non unital C*-algebra that has an element with empty spectrum, if it exists. Motivation I ...
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125 views

Spectral radius of an element in a C*-algebra

The following is an proposition of Takesaki's Operator Theory: For any element $x$ of a Banach algebra ${\cal A}$, we have $$||x||_{sp}=\lim_{n\to \infty}||x^n||^{\frac{1}{n}}$$ Proof: My ...
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34 views

What operation is being done for this set of values?

I have a table that looks like the following: A B C A | B A C B | A C A C | C A B Some operation is being done between an element in the ...
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59 views

Why is the weak operator closure of a commutative $\boldsymbol{C^*\!\!\!\!-}$algebra also commutative?

In a book on Operator Theory there is the following statement: If $\mathscr A$ is a commutative $C^*$-subalgebra of $\mathscr B(\mathcal H)$, where $\mathcal H$ is a Hilbert space, then the weak ...
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1answer
26 views

Continuity of an integral operator

I'm stuck with this exercise: Let $A \subset \mathbb{R}$ be a measurable set. For each $f \in L^1(\mathbb{R})$ and $y \in \mathbb{R}$, let: $T(f, y) = \int_{A}{f(x-y)\mathrm{d}x}$. I have to show ...
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65 views

Does the integral operator, whose kernel is the indicator of the rhombus, belong to the trace class?

In connection with this question: Does the integral operator on $L^2(\mathbb R)$, whose kernel is the indicator of the rhombus $\{|x|+|y|<1\}$, belong to the trace class?
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111 views

What happens if we rotate the kernel of an integral operator?

Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, ...
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46 views

Does an operator of x commute with the differential operator with respect to x?

While solving a problem in Quantum Mechanics I got an expression $ \frac{d}{dx}V(x)-V(x)\frac{d}{dx} $. The first term is just the derivative of the potential but the second one seems a bit weird. Is ...
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152 views

Proof of the product rule for the divergence

How can I prove that $\nabla \cdot (fv) = \nabla f \cdot v + f\nabla \cdot v,$ where $v$ is a vector field and $f$ a scalar valued function? Many thanks for the help!
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35 views

Those differential operators that are bounded.

Differential operators are known as unbounded operators, but there always are some exceptions. Does anyone know an example of a differential operator on appropriate Sobolev spaces that is not ...
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54 views

Bounded operators that are not closed.

If a bounded operator, say $A:D(A)\to X$, have $D(A)=X$ then it is closed. Can anybody construct an example of a bounded linear operator, without resorting and restricting to $D(A)=X$, that is not ...
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46 views

Notation of measures $d \mu$

I am reading the paper http://www.ams.org/mathscinet-getitem?mr=3246935 and there some notation that I have found a bit confusing on page 1503 between Lemma 4.2 and 4.3. I'll give as much context as I ...
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25 views

When a symmetric densely defined operator is an adjoint operator?

I am wondering if it is possible to say that if a symmetric differential operator is densely defined then the operator is self-adjoint indeed? More Precisely, Let $A:D(A)(\subset H)\to H$ a densely ...
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48 views

Closed Operator on a Sobolev space

I am wondering if the following differential operator $A:D(A)( \subset {\bf{H}}) \to {\bf{H}}$ defined on the sobolev space $\mathbf{H}=H_{0}^{k}(0,1)\times {{L}^{2}}(0,1)\text{ }$ is a closed ...
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54 views

Proof of distributive property of linear operator?

How can I show that for 2 linear operators $L$ and $M$ that transform some object $O$ into another object of the same type: $$(L(O)+M(O))*(L(O)+M(O)) = L(O)*L(O) + 2L(O)*M(O) + M(O)*M(O)$$ where $*$ ...
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1answer
35 views

Norm of a character in a non-unital Banach algebra

Let $\cal A$ be an abelian non-unital Banach algebra and $h:{\cal A}\to {\Bbb C}$ be a homomorphism. If ${\cal A}$ has an approximate identity $\{e_i\}$ such that $||e_i||\leq 1$ for all i, then ...
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2answers
70 views

spectrum of operators between normed spaces

The spectrum of a linear operator $L: \mathcal{D}(L) \rightarrow \mathcal{X} $ is generally defined for $\mathcal{X}$ a Banach space (for example wikipedia on link above, or spectral decomposition on ...
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80 views

Finding the norm of this upper triangular matrix

I have a matrix $A=\begin{pmatrix} a & b\\ 0 & a\end{pmatrix}\in M_2(\mathbb{C})$. Given that $|a|<1$ and $|b|\leq 1-|a|^2$, I am supposed to show that $\|A\|\leq 1$ (operator norm). I ...
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1answer
34 views

When Heine - Borel theorem holds

If $\cal A$ is an abelian Banach algebra, then its maximal ideal space $\Omega$ is a compact Hausdorff space. In the proof of this theorem, the author says, since $\Omega \subset ball \cal A^*$, it ...
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105 views

isometric isomorphism between normed spaces and its dual

Let $E$ and $F$ be normed spaces. If $E \equiv F$ (isometry isomorphic), Does $E^* \equiv F^*$ (isometry isomorphic)? Where $E^*$ and $F^*$ are continuous dual spaces.
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44 views

Every homomorphism on a C*-algebra is a *-homomorphism

The following is a proposition of Conway's Functional analysis: and also he uses below exercise to proof above proposition: But I do not know how he uses the Exercise and say $||h||=1$. Please ...
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28 views

Is range a of a generator of a strongly continuous semi group in doman of the generator?

Let $X$ be a banach space and $A:D(A)\rightarrow X$ be a generator of a infinte seminal generator of a $C_0$ semi group $\{S(t)\}_{t\geq 0}$. In this case is it possible that $Range(A)\subset D(A)$. ...
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1answer
60 views

Example of a proper, dense ideal in a unital Banach algebra

I know that if ${\cal A}$ is a non- unital Banach algebra, then ${\cal A}$ may contain some proper, dense ideals. I need an example of a proper, dense ideal in a unital Banach algebra or show that it ...
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1answer
40 views

Sign of the eigenvalues of the Laplacian

I have to prove that, given the problem$$ \begin{cases} \Delta\:g+ \lambda \:g=0\quad {\rm in}\;D \\ g=0\quad {\rm on} \; D\end{cases}$$ then the eigevalues $\lambda>0$. I multiply the first ...
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1answer
84 views

The spectral theory of unbounded operators

I would like to learn about the spectral theory of unbounded operators. I'm looking for a lucid, rigorous, self-contained and basic exposition of this topic that assumes no more than the material ...
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22 views

“Interpolating between estimates”?

the headline reproduces the whole problem. What is meant by saying "Interpolating between the estimates (A) and (B), we finally obtain..."? For beeing mor specific I'll give the concrete estimates ...
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1answer
53 views

Hilbert Spaces - an application of the minimax principle.

Let $A$ be a compact, self-adjoint operator, $A \geq 0$. We need to prove that for any orthonormal system $\{e_i\}_1^{\infty}$ and for any $N$, $$\sum_1^N \langle Ae_i,e_i \rangle \leq \sum_1^N ...
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1answer
65 views

Operator norm equality

I came across this problem and am getting stuck on how to prove it. Any help would be appreciated. Suppose $L:C(\textbf{T}) \rightarrow \mathbb{C}$, where $L(f)=\int_0^1 {f(x)g(x)}dx$ for all $f \in ...
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1answer
17 views

Sum of products of positive operators

I'm trying to answer the following question: Given two positive self adjoint operators $\mathcal{A}$ and $\mathcal{P}$ on a Hilbert space, is the following composition: $\mathcal{AP}+\mathcal{PA}$ ...
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1answer
119 views

Is an orthogonal projector bounded in $L_p$-spaces?

Let $P$ be an orthogonal projector on $C^\infty([0,1])$. For $0<p<\infty$, we define for $f \in C^\infty$ the norm (quasi-norm if $p<1$) $\lVert f \rVert_p$ in the usual way. Moreover, we ...