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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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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23 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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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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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
29 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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66 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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1answer
63 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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29 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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90 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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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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27 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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48 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
39 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
77 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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21 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
42 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
62 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
14 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
118 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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1answer
44 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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89 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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1answer
55 views

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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40 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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1answer
34 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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71 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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1answer
22 views

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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55 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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70 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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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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24 views

Relation between two spectra

This seems like an easy enough computation but I'm stuck! Let $X \in B(H)$ for a Hilbert space $H$ such that $X^{2}=0$, but $X\neq 0$. With respect to the decomposition $H=\text{ker}X \oplus ...
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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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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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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: ...
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38 views

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 ...
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62 views

bounded linear operators is B(X,Y) is complete if Y is complete

QUESTION#1 is why he required that the space Y is complete not only the range of the operators is complete? QUESTION#2 In the proof of this theorem : i take a cauchy sequence {Tn} from B(X,Y) and ...
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107 views

Question about proof of Browder, Minty Theorem

Could someone please assist with the following question: In the following SET OF NOTES, I am interested to know how the author obtains "By Lemma 1.11, the Galerkin equations (2.5 has a solution ...
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32 views

How to show an operator is invertible?

How to show an operator is invertible in an abstact setting? I only know that if $\|T\|< 1$ then $I-T$ is invertible. For example: Let $(X_1, \|\cdot\|_1)$ and $(X_2, \|\cdot\|_2)$ be Banach ...
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28 views

Intersection of $\ell_p$

Let $\ell_p$ be defined as usual then what can be said about the intersection of $\ell_p$ spaces for i.e. $$\,\,\cap_{p=1}^{\infty}\ell_p\,\,$$
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Given $A_n : X\rightarrow Y$ linear and continuous, for each $x\in X$ $A_n(x) \rightharpoonup A(x)$ in $Y$, is $A$ continuous?

Given $A_n : X\rightarrow B$, a linear and continuous operator between two Banach spaces, for each $x\in X$ $A_n(x) \rightharpoonup A(x)$ in $Y$, is $A$ linear and continuous? My attempt: $A$ ...
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2answers
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Bounded linear operator commuting with every compact operators

Let $A$ be a bounded linear operator on the Banach space $X$. Assuming that $AK = KA$ for every compact operator $K$, how do I show that $A$ must be a scalar multiple of the identity, i.e., we have $A ...
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1answer
72 views

Irreducible representations and commutative C*Algebras.

If $A$ is a commutative C*-Algebra then also its representation $\pi(A)$ is commutative, and it's an operator C*-algebra. A representation is said to be irreducible if $\pi(A)$ does not commute with ...
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30 views

Solution of nonhomogeneous problem using semigroup of linear operators

Let $X$ be a complex Hilbert space; and let $A:D(A)\subset X \to X$ be a $\mathbb C-$linear operator. Assume that $A$ is self-adjoint(so that $D(A)$ is a dense subset of $X$) and that $A\leq 0$ (i.e., ...
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1answer
20 views

Perturbation of operators and eigenvalues

Suppose $P\in\mathcal{B}(\mathcal{H})$ is a self-adjoint compact operator. Lets perturb $P$ by multiplying it by a bounded operator $S$ and set $T=PS.$ Then what can be said for the spectrum of $T?$ ...
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functional analysis: show L^1 integral operator has norm 1

I just started my course in functional analysis and have already stumbled across some things I don't understand, which are quite basic :(. In my lecture notes it says: Let $\mu$ be a measure on a ...
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215 views

Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
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37 views

Strong closure of a C*-algebra of operators.

In Arveson's book, the Kaplansky density theorem is proved in order to have this corollary: "Let $A$ be a self-adjoint algebra of operators on a separable Hilbert space $H$. Then for every operator ...
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41 views

Prove that $\frac{1}{\phi(D)}c=c\frac{1}{\phi(0)}$

Could someone prove the following? $$\frac{1}{\phi(D)}c=c\frac{1}{\phi(0)}$$ where $D$ is ${\frac{d}{dx}}$ and $c$ is a constant. for example $$\frac{1}{D^4+2D+3}c=c\frac{1}{0+0+3}=\frac{c}{3}$$