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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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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69 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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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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80 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 ||x||_{...
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119 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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37 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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280 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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291 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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39 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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94 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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73 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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118 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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77 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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2k 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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122 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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176 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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80 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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54 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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117 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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87 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 $||h||...
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Why is the spectrum usually defined for operators between Banach spaces?

The spectrum of a linear operator $L: \mathcal{D}(L) \rightarrow \mathcal{X} $ is generally defined for $\mathcal{X}$ a Banach space (as seen for example wikipedia on link above, or spectral ...
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375 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 can'...
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77 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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372 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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Is the range a of a generator of a strongly continuous semigroup contained in the domain of the generator?

Let $X$ be a banach space and $A:D(A)\rightarrow X$ be a infinteseminal generator of a a $C_0$ semi group $\{S(t)\}_{t\geq 0}$. In this case is it possible that $\operatorname{Range}(A)\subset D(A)$?...
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112 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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51 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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105 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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24 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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99 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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168 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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45 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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129 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 ...
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85 views

Hilbert Spaces; eigenvalues of $PBP$ vs. $B$ for $B$ compact selfadjoint and $P$ orthoprojection.

An exercise I have come upon while studying Hilbert Spaces: Let $A$ be a compact operator, and $P \in L(H)$ be an orthoprojection. Prove that $$\lambda_n (PA^*AP) \leq \lambda_n (A^*A)$$ (Where $\...
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241 views

What is the general form of linear operators on continuous functions?

I was wondering if there was a representation for a set of operators dense in the space of linear operators $B$ mapping $C(a,b) \to C(c,d)$. I thought that maybe integral operators give a general ...
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Why are “not bounded” operators not everywhere defined?

Let $X, Y$ be Banach spaces, $\mathcal{D}(T)$ a subspace of $X$, and $T\colon X\to Y$ a linear map. Such a $T$ is commonly called an unbounded linear operator, where unbounded just means that the ...
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51 views

An exercise about the positive operator

Here is an exercise in functional analysis: An operator $T$ on Hilbert space is positive is positive if and only if all compressions by finite-rank projections ($P_{n}TP_{n}$ for any $n$) are ...
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When two projections in a C*-algebra are “almost” Murray-von Neumann equivalent, they are equivalent

Let $A$ be a C*-algebra and $p,q \in A$ be projections. Assume there is an element $a\in A$ such that $\|aa^*-p\|<\frac{1}{4}$ and $\|a^*a-q\|<\frac{1}{4}$. Then there is a partial isometry $v$ ...
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Show that $T:X\rightarrow X$ is chaotic iff every finite family of nonempty open sets shares a periodic orbit.

Here is a problem from Grosse-Erdmann and Peris' Linear Chaos book that I am trying to solve. Exercise 1.3.4. Show that $T:X\rightarrow X$ is chaotic iff every finite family of nonempty open sets ...
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103 views

Continuous spectral theorem example

The spectral theorem can be explicitly expressed for an hermitian matrix by providing its eigen decomposition. In the more general case of a bounded self-adjoint operator with a continuous spectrum, ...
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143 views

Exercise 23 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 23 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 198). Any help will be much appreciated. Thank you in advance. Suppose $\{T_k\}$ is a collection of bounded ...
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Extending mappings on simple tensors

Consider the following situation: Let $H, K$ be Hilbert spaces and let $\Phi$ be some mapping defined on simple tensors in $H\otimes K$ taking values in $B(H\otimes K)$ with the property that each ...
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107 views

Basis of eigenfunctions in Banach space

I have a question about proving the existence of a basis of eigenfunctions. Assume we have a compact operator $L:\mathcal{B}\rightarrow\mathcal{B}$, where $\mathcal{B}$ is a Banach space of analytic ...
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307 views

Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

The following problem shows that $\{e^{inx}\}_{n \in \mathbb{Z}}$ is an orthonormal basis of $L^2([-\pi, \pi])$. It is taken from [1] (exercise 31 of chapter 4: "Hilbert Spaces: An Introduction", pp. ...
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398 views

Exercise 34 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 34 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 201): Let K be a Hilbert-Schmidt kernel which is real and symmetric. Then, as we saw, the operator $T$ whose ...
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Problem 8 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein and Shakarchi's Real Analysis

The following is problem 8 from chapter 4 ("Hilbert Spaces: An Introduction") of Stein and Shakarchi's Real Analysis. Suppose $\{t_k\}$ is a collection of bounded operators on a Hilbert space $H$. ...
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81 views

Prove that T is compact

If $H$ is a Hilbert space with basis $\{\varphi_{k}\}^{\infty}_{k=1}$, how do I show that the operator $T$ defined by $T(\varphi_{k})=\frac{1}{k}\varphi_{k+1}$ is compact and has no eigenvectors? ...
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48 views

Compact Operator on Hilbert Space

How do I show that the range of $\lambda I-T$ is all of $H$ (Hilbert Space) if and only if the null-space $\bar\lambda I-T^{\ast}$ is trivial? Thanks!
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101 views

Quantum translation operator

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here's what I've gotten: $$T_\...
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Operator norm and Hilbert Schmidt norm

I'm looking for a proof of \begin{equation} ||T||\leq ||T||_{HS}, \end{equation} for which it is sufficient to show \begin{equation} ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 \end{...