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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Equivalent Definitions of the Operator Norm

Would you give me a proof of the equivalence of these ones? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ ...
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0answers
12 views

Daletskii-S.Krein formula proof

I've came across to the following equation, known as Daletskii-S.Krein formula. Consider a sufficiently smooth function $h : \mathbb{R} \rightarrow \mathbb{R}$, and let $\mathbf{A}_t = \mathbf{A} + ...
5
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1answer
459 views

Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?

Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem Does the following generalization of that fact also hold? Theorem: ...
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2answers
135 views

Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentially selfadjoint if it contains a dense subset of analytic vectors?
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1answer
37 views

If a linear operator between two normed linear spaces is continuous at one point, then it is continuous at all points.

Let $f : \langle V_1, \|\cdot\|_1\rangle \to \langle V_2, \|\cdot\|_2\rangle$ be linear. Then if $f$ is continuous at some $v \in V_1$, then it is continuous on all of $V_1$. Without appealing to ...
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2answers
38 views

The norm of a bounded linear operator has this formula: $\|T\| = \sup_{\|v\| = 1} \|T v\|$

Trying to prove $\|T\| = \sup_{\|v\| = 1} \|T v\|$, given $\|T\| := \inf_{C \geq 0} \{C: \|Tv\| \leq C\|v\|\}$. I know that $\|T(v)\| = \|T(\alpha \hat{v})\| \leq C\|\alpha \hat{v}\|$ for $v = ...
3
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2answers
67 views

Approximate Identities

Statement : Let $U$ be a C* algebra and$\lambda=\left\{A\in U:A\geq 0, ||A||<1\right\}$. If $B\in \lambda$ then if $X\in U$ $$||X^*(I-B)^2X||\leq ||X^*(I-B)X||$$ For reference this is from ...
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2answers
50 views

Volterra-like operator is bounded

Define $T:L^2(\mathbb R) \rightarrow L^2(\mathbb R)$ by $$(Tf)(x)=\int_{-\infty}^x e^{-(x-y)} f(y) \, dy.$$ I would like to show that $T$ is bounded and that $$\lambda = \frac{1}{1+iw}$$ is in its ...
0
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1answer
35 views

Is there a name for operators of the type $A: M \to M$

In some theorem in functional analysis I have noticed that it is important to assume that an operator $A: M \to M$ where $M$ is some set plus conditions, as opposed to $A: M \to N, M \neq N$ Is ...
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1answer
25 views

significance and importance of spectral theorem

I have started recently started Operator Theory and have been introduced to the Spectral Mapping Theorem: If $a \in \mathcal{A}$, where $\mathcal{A}$ is a unital Banach Algebra and $f \in ...
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1answer
39 views

Definition in Operator Theory

I have started learning some Operator Theory. I encountered the following definition. I would like to know why it is that the $f(z)$ in the integrand and the $f(a)$ are both labelled as $f$ where it ...
0
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1answer
67 views

Projections: Beppo Levi

Given a Hilbert space $\mathcal{H}$. Consider projections: $$P_\lambda\in\mathcal{B}(\mathcal{H}):\quad P_\lambda^2=P_\lambda=P_\lambda^*$$ And directed indices: ...
0
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0answers
19 views

approximate unit of $K(H)$- ordering on $K(H)$ and finite rank operators

Let $H$ be a complex Hilbert space with orthonormal basis $\{e_i:i\in I\}$ . Consider the $C^\ast$-algebra of the compact operators on $H$, $K(H)$. For a finite subset $F\subseteq I$, let $P_F$ be the ...
0
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0answers
15 views

For what does the formula $(\prod_{t=1}^d[\begin{array}{c}-\frac 12&1&-\frac 12\end{array}]_{l_t,i_t})f$ stand for?

Let $f:\mathbb R\to\mathbb R$ and $$a_{l,i}:=f(x_{l,i})-\frac{f(x_{l-1,(i-1)/2}+f(x_{l-1,(i+1)/2})}2$$ for some $x_{l,i}$. I've read, that we can write $a_{l,i}$ in the following "operator form": ...
4
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2answers
88 views

Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
1
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1answer
18 views

Continuous spectrum is a subset of point spectrum

I have to prove that the continuous spectrum $\sigma_c(T)$ is a subset of the point spectrum $\sigma_p(T)$. I started off by supposing that there is some spectral value $\lambda$ such that $\lambda ...
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2answers
72 views

Eigenvectors of operators on a tensor product Hilbert Space

Suppose I have finite dimensional Hilbert spaces $V$, $W$, and an operator $A$ acting on vectors in $V$ such that it has eigenvectors/values $Ax_a=\lambda_ax_a$. In the tensor product space I want to ...
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1answer
16 views

Invariant subspace and projection

Let $F$ be a subspace of a Hilber space $H$, invariant under a bounded linear map $T$, and let $P$ be an orthogonal projection such that $Im(P)=F$. I need to show that $F$ and $F^\perp$ are ...
2
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2answers
30 views

Operator Norm ( Confusion)

I am reading a book about operator theory and it states the following; If $X$ and $Y$ are two normed spaces and $T:X\rightarrow Y$ is an operator, define it's norm by; $$\|T(x)|| = \sup \{ \| T(x) ...
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1answer
31 views

Proof that every polilinear map who's domain is $R^{n_1} \times R^{n_2}… \times R^{n_k}$ and co-domain any given real normed space Y is bound.

A Polilinear map\operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
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0answers
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3
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1answer
48 views

$f\mapsto \frac{df}{dx} - \frac{x}{\sqrt{1+x^2}}f $ has closed image and $1$-dimensional cokernel

Let $X$ be the completion of the space of smooth, compactly supported real-valued functions on $\mathbb R$ under the norm $$\|f\|_X^2=\int_{\mathbb R} \left(\frac{df}{dx}\right)^2 + f^2.$$ Let ...
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0answers
26 views

Jordan normal form

Let $H$ a Hilbert space and let $T\in B(H)$ a bounded operator on H, my question is if it exist a theorem about some "decomposition" of type Jordan canonical form in a general Hilbert space, and how ...
6
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3answers
192 views

Bounded operators with prescribed range - part II

This is a continuation of the question bellow, in a more particular case. Bounded operators with prescribed range If $X$ is a separable Banach space and $Y$ is a closed, infinite dimensional ...
2
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1answer
57 views

Power series expansion of an Operator.

I've been reading a paper called "Separation of variables for the quantum $Sl(2,R)$ spin chain" in which the author at one point does a power series expansion I do not understand. The problem is this ...
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2answers
285 views

Composition of Fredholm Operators

If $ST$ is a Fredholm operator, then show that $T$ is Fredholm if and only if $S$ is Fredholm.
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1answer
60 views

Fredholm index of invertible bounded operator

Let $X$ be a Banach space and $T: X \to X$ be bounded and invertible. Is it true that the Fredholm index $\mathrm{ind}(T) = 0$?
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1answer
33 views

Dense domain of Unbounded Operator

Let $H$ a Hilbert space and $A:D(A)\subsetneq H\rightarrow H$ a dissipative, unbounded linear operator with $R(A)=Im(A)$ closed in $H$, such that exist $A^{-1}$, bounded linear operator. How I can ...
2
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1answer
58 views

WOT convergence in the unit ball of B(X)

My questions is (probably) related to: On separable Hilbert space $H$, weak operator topology is metrizable on bounded parts of $B(H)$ Does the theorem quoted in the above question, together with ...
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1answer
46 views

Extending finite rank operators

Suppose $Y$ is a closed subspace of Banach space $X$ and $T:Y\to X$ is a bounded finite rank operator. Can we extend $T$ to $\tilde{T}:X\to X$, in the sense that: $T=\tilde{T}$ on $Y$ ...
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0answers
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Find the iverse of the followning bounded operator?

The following definition and Theorem are given in the book "A short course on operator semigroup" by the author "K-J Engel and R Nagel". Sectoral operator: A closed linear operator $(A,D(A))$ in ...
2
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0answers
41 views

Construct an operator that fixes the equivalence class of Cauchy sequences

Let $X$ be a Banach space and $\overline{X}$ be its unique completion. We know that $\overline{X}$ can be partitioned into equivalence classes of Cauchy sequences via the relation $\sim$: $$ \{x_n\} ...
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1answer
76 views

Showing the compactness of a limit operator.

I was trying to solve this exercise from Kreyszig's book, section 8.1 exercise number 10. My attempt was try to show that the operators in the sequence are bounded, but I don't find it. If this fact ...
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1answer
27 views

Spectral Measures: Multi Version (II)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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1answer
37 views

Spectral Measures: Multi Version (I)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad ...
0
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1answer
28 views

Show for the Hamilton's operator $H$ that $\overline{(H, C_0^{\infty}(\mathbb{R}))} = (H, W_2^2(\mathbb{R}))$ using Fourier transform

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable real-valued function defined on $\mathbb{R}$ bounded with its first derivative. Consider the Hamilton's operator $H$ such that: ...
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1answer
67 views

can i prove or reject it?

Let $\Phi$ be a surjective map on an algebra $\mathcal{A}$ which satisfies the following condition for a fixed arbitrary $\epsilon \in \mathbb{C}$ which $\epsilon\neq1,-1$ and for fixed ...
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2answers
66 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 ...
4
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1answer
59 views

is the pullback operator associated to a flow bounded in L^2?

Let $M$ be a smooth compact manifold with a finite Borel measure $m$. Let $\{f_t\}_{t\in\mathbb R}$ be a $C^1$ flow on $M$. That is, a $C^1$ function $$ \mathbb R\times M\ni(t,x)\mapsto f_t(x)\in M $$ ...
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2answers
30 views

Hahn Banach Theorem: Clarification on meaning of extending a functional?

Hahn Banach Theorem: Given linear (vector) space $\mathbb{X}$, define $u \in \mathbb{L} \subset \mathbb{X}$, $A,B,C$ functionals, A sublinear. $A:\mathbb{L} \to \mathbb{R}, B:\mathbb{L} \to ...
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1answer
77 views

Is there a $C^{*}$ algebra with these properties

Is there a unital C* algebra A which is NOT simple but satisfies the following two conditions: 1)A has trivial center 2)A has a faithful trace such that every zero trace element lies in the closure ...
2
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2answers
68 views

Are all matrices linear operators?

Given $A \in \mathbb{K}^{n\times m}$ a matrix, can we think of $A$ as an operator? In what context do matrices satisfy the definition of operator?
2
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1answer
28 views

Partial Isometries: Final

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$J\in\mathcal{B}(\mathcal{H},\mathcal{K}):\quad P:=J^*J$$ By the C*-property: $$J=JJ^*J\iff P^2=P=P^*$$ Note that in any ...
4
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1answer
47 views

Taylor expansion of $\frac{1}{1-D}$, where $D$ is the differential operator

I understand that we can represent $e^D$ simply as a power series of D. But what about functions of D which are not entire on the complex plane? What if the function has no taylor expansion, or if it ...
8
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4answers
708 views

is $T:C[0,1]\rightarrow C[0,1]$: $x(t)\mapsto x(t^2)$ compact?

Is $T$ a compact operator? $T:C[0,1]\rightarrow C[0,1]$: $x(t)\mapsto x(t^2)$ where $t\in[0,1]$ with supremum norm.
2
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1answer
25 views

(Operator) norm inequality for continuous functions

Let $f,g$ be two non-negative continuous functions on $[0,\infty)$ satisfying $f(t)g(t)=t,$ $\forall t\in[0,\infty)$. Let be $A$ be a bounded linear operator acting on a Hilbert space. Then I was ...
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2answers
53 views

Weighted shift operator is Hilbert-Schmidt

If $W : \ell^2 \to \ell^2$ is the weighted shift operator defined by $$W(x_1,x_2,x_3,\ldots)=(0,x_1,\frac 12x_2,\frac 13x_3,\ldots),$$ how can I show that $W$ is Hilbert-Schmidt? If I have ...
0
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1answer
71 views

Partial Isometries: Subspaces

This thread was only Q&A. Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$J\in\mathcal{B}(\mathcal{H},\mathcal{K}):\quad P:=J^*J$$ By a previous thread:* ...
3
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1answer
59 views

Solution to Equation $Ax=f$ in Hilbert Space

Question. Let $H$ be a separable Hilbert space with complete orthonormal basis $\left\{u_{k}\right\}_{k=1}^{\infty}$, let $H_{n}:=\text{span}\left\{u_{1},\ldots,u_{n}\right\}$, and let ...
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1answer
22 views

A question on Operator of a Banach Space

For any $x \in X$ where $X$ is a Banach space, is there a non-trivial bounded operator $T \in B(X)$ such that $T(x)=x$? I mean is there any way to verify the existence of such an operator for any $x ...