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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minimal projections in matrix-algebras

Consider $A=\{ \begin{pmatrix} T & 0 \\ 0 & T \end{pmatrix}: T\in M_2(\mathbb{C})\}\subseteq M_4(\mathbb{C})$ and $p= \begin{pmatrix} 1 & 0&0&0 \\ 0 & 0&0&0\\0 & ...
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Polar decomposition theorem for symplectic and orthogonal Banach Lie groups in infinite dimensional settings

Could you please help me to understand the polar decomposition theorem for $Sp(H, J_Q)$ and $O(H,J_R)$ where $H$ is infinite dimensional separable Hilbert space and $J_R$ and $J_Q$ stands for ...
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17 views

Weakly convergent subsequence under continuous operator

Suppose we have two Hilbert spaces $H_1,~ H_2$, a linear continuous operator $T:H_1 \to H_2$ and a weakly convergent sequence $u_k\rightharpoonup u$ in $H_1$. Is $Tu_k \rightharpoonup Tu$ in $H_2$ ...
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13 views

Bounding a linear map from $L^q$ to $C_c$

Let $C_0((0, \infty))$ be the set of all functions such that $\lim_{x \to \infty} f(x) = 0$ and $\lim_{x \to 0} f(x) = 0$, this is a Banach space under the $\sup$ norm. Now if we fix a $p$, $1 < ...
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difference of projections is a positive operator

Let $p,q:H\to H$ bounded, linear operator on a Hilbert space $H$, such that $p^*=p=p^2$ and $q^*=q=q^2$ ($p^*$ is the adjoint of $p$, for q the same. YOu call $p$ and $q$ a projection). Let ...
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proving equivalence of strongly continuity

A semigroup $S(t)$ on a Banach space $E$ is a family of bounded linear operators $\{S(t)\}_{t\ge 0}$ with the property that $S(t)S(s)=S(t+s)$ for any $s,t\ge 0$ and that $S(0)=I$. A semigroup is ...
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27 views

(bounded linear) orthogonal projections on Hilbert spaces

If $H$ is a Hilbert space and $T:H\to H$ bounded and linear, such that $T$ is an othogonal projection (i.e. $T^*=T^2=T$), is then T always zero on $im(T)^\perp$ and the identity on it's image, ...
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12 views

Name of operators somewhat similar to differential operators returning “pace” of functions

I have a set of operators with specific properties, and I believe that somebody must have studied (and baptized) them before. The operators remind me of differential operators, however-as far as I ...
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24 views

Substitution from the left and right?

I am a bit confused about this substitution from the left or from the right thing. How does one determine whether to substitute from the left or from the right? And does it even matter as long as i ...
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28 views

Inverse of a dissipative operator

In the spectral theory of linear operators it is often helpful in the development of proofs for many results to proceed with the inverse of a (boundedly invertible) dissipative operator. For example, ...
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82 views

Show strong but not norm convergence on $L^p$

The task: let $1 \le p < \infty$ and $A_k : L^p (\Bbb R) \to L^p (\Bbb R)$ such that $(A_k u) (x) = u(x+\frac 1 k)$. Show that $\| A_k u - u\|_p \to 0$ as $k \to \infty$ (for all $u \in L^p (\Bbb ...
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The subgroup of $GL_{n}(\mathbb{R})$ inside $B(1, 1)$ is $\lbrace I \rbrace$

Let $G \subset GL_{n}(\mathbb{R})$ be a subgroup such that $G \subset B$ where $$B = \lbrace M \in \mathcal{M}_{n}(\mathbb{R}), ||M-I|| < 1\rbrace $$ Let $g \in G$. I'm asked to show: The ...
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36 views

Irreducible representation

Suppose $H$ is a separable Hilbert space, and $K$ is a Hilbert- schmidt space on $H$. We know $K$ is a Hilbert space. Consider representation $\pi : B(H) \to B(K)$ such that $\pi(a)x:= ax$. Proveing ...
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49 views

Eigenvalues of adjoint operator [General Case]

I am asked to prove that if $T: V \rightarrow V$ is a linear operator over a complex inner product space $(V,\langle,\rangle)$, then $\overline{\lambda}$ is an eigenvalue of $T^*$ where $\lambda$ is ...
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66 views

Arzelà–Ascoli theorem for operators

If you have a net of continuous linear operators between reasonable spaces (complete, at least), does there exist Arzelà–Ascoli-like theorems giving convergent subnets? I believe that it should be ...
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Characterization of normal operators on Hilbert space as function of a self-adjoint operator

My question : Suppose T is a normal operator on a Hilbert space H. Show that there exists a self-adjoint operator S on H such that T=f(S), where f is continuous function from spectrum of S into S. My ...
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68 views

Using Fubini's Theorem in Contour Integrals proof

I have a few questions regarding the following proof: Suppose that $\mathcal{A}$ is a unital Banach algebra, and that $g$ is a complex-valued function which is analytic on $\sigma(a)$ while $f$ is a ...
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22 views

Absolut convergence implies being in trace class

I'm asking for your help in the next problem, I can't think how to do it! Prove that if $$\sum_{n=1}^{\infty}|(A\phi_n,\phi_n)|<\infty$$ for all orthonormal bases, then $A$ is in the trace class. ...
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97 views

Simplifying to Linear Differential Operator?

Days ago and I'm trying to understand this equation please help If the linear differential operator $$C= C_1 \frac{\partial}{\partial x}+C_2 \frac{\partial}{\partial y}+C_0$$ and $\phi(x,y)$ ...
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Can $AB-BA=I$ hold if $A$ and $B$ are bounded linear operators on a Banach space?

It's easy to see it can't be true in finite dimension. Also, it can hold for operators on infinite-dimensional vector spaces over $\mathbb C$, as seen here: Can $AB-BA=I$ hold if both $A$ and $B$ are ...
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43 views

Proof involving invertible elements of Banach algebra

I want to prove for a unital Banach algebra $\mathcal{A}$, it follows that if $\|a-b \| < \frac{1}{\|a^{-1} \|}$ then $b \in \mathcal{A}^{-1}$ (where $\mathcal{A}^{-1}$ is the subset of invertible ...
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Proposed proof of operator theory result

Hi I am interested in checking my proposed solution to the following problem in Operator Theory: Please give me hints as to how to improve the proposed proof rather than the full correct solution. ...
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22 views

Uniform boundedness of a family of bounded operators

Suppose that the map $\lambda\to T(\lambda)$ from the domain $\{0<|\lambda|\leq 1\}\subset \mathbb{C}$ to the space of bounded operators on a Banach space $B$, . And further suppose that $T(0)$ is ...
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Introductory books for ‎ ‎$\frak{E}_p(I)$

Are there any good books different from abstract harmonic analysis by hewitt to study ‎$\frak{E}_p(I)$. where ‎$\frak{E}_p(I)$ is: ‎Let $I$ be an arbitrary index set‎. ‎For each $i\in I$ let $H_i$ ...
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24 views

Operator theory and error function bound

Hi guys i have another question about operator theory, Could operator theory potentially help in finding error bound for an approximation? I make an example taken from classical result in numerical ...
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38 views

Sobolev “unit ball” compact in $L^p$

We consider $I=(0,1)$, and $1< p \leq \infty$. If $B = \{ f \in W^{1,p}(0,1); ||f||_p + ||f'||_p \leq 1 \}$, how to show that $B$ is compact in $L^p(0,1)$?
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34 views

Matrix representatin of a compact operator

Fact : I know for every compact operator $x : H \to H$, there are sequences $\{\xi_n\}$ and $\{\eta_n\}$ of orthonormal vectors of separable Hilbert space $H$, and sequence $\{\alpha_n\}$ in $\Bbb C$ ...
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20 views

Question about norm in trace class

I'm having troubles to proof the next inequality $$\|A\|\leq\|A\|_1$$ where A is an operator in the trace class and $\|A\|_1=Tr|A|$. And $\|A\|$ is the norm of operators. I just got this ...
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Proving that an operator $T$ on a Hilbert space is compact

Let $H$ be a Hilbert space, $T:H \to H$ be a bounded linear operator and $T^{*}$ be the Hilbert Adjoint operator of $T$. Show that $T$ is compact if and only if $T^{*}T$ is compact. My attempt: ...
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53 views

Existence of a surjective compact linear operator on an infinite dimensional Banach space

Does there exist a surjective compact operator $T:l^{\infty} \to l^{\infty}$ ? Even though this might be tagged as a repeat question, i still have some doubts that i would like to clarify. My ...
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35 views

Proving that an operator is Compact

I have to check that the following operator $T$ is compact: Define $T:l^{2} \to {l^2}$ by $Tx=y=(\eta_{j})$, where $x=(\zeta_{j})$ and $(\eta_{j})=\sum_{k=1}^\infty \alpha_{jk} \zeta_{j}$ and ...
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Proving that Eigenvectors Span Hilbert Space

I have a specific problem I am trying to solve, but I would like to learn general principles, so I will start my question pretty general and add specifics later. Please answer the most general form of ...
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Compactness of a linear operator

The question is as follows: Show that a linear operator $T:X\to Y$ where $X$ and $Y$ are normed spaces is compact if and only if the image $T(M)$ of the unit ball $M\subset X$ is relatively compact ...
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39 views

Spectrum of compact operators on an infinite dimensional normed space

The question is as follows: Let $T:X \to X$ be a compact linear operator on a normed space. If the $dim X= \infty$ then show that $0 \in \sigma(T)$. My attempt: Suppose on the contrary that $0 ...
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checking definition of bounded linear function involves operator maps between different spaces

Let $H$ and $K$ be two Hilbert spaces. Let $T:K\to H$ be a bounded linear operator. Denote the inner products on $H$ and $K$ by $\langle\cdot,\cdot\rangle_H$, $\langle\cdot,\cdot\rangle_K$. Fix any ...
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54 views

Is there any inner product on $M_{n \times n}$ inducing this norm?

The set $M_{n \times n}$ is the collection of all $n \times n$ matrices over $\mathbb{R}$. Definition: $\|A\|_2=\sup_{\|u\|_2=1} \|Au\|_2$. Is there any inner product on $M_{n \times n}$ inducing ...
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26 views

Verification of some conditions and facts of the Laplacian on a Riemannian manifold

I came across the following introduction to a paper I was reading: "Let $L$ be a second-order, elliptic, differential operator, with smooth coefficients, on a Riemannian manifold $M$. Assume $-L$ is ...
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25 views

If $A: M \to M$ then $M$ is $A$-invariant subspace and $A $ is an endomorphism?

Just straightening out the terminologies here... Given If $A: M \to M$ then $M$, $M$ some subspace of a vector space, is the following statement equivalent: $M$ is a $A$-invariant subspace $A $is ...
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99 views

How numerical radius help us to conclude an operator is normal and partial isometry?

In Furuta's book, "Invitation to Linear Operators" there is a theorem, theorem 2 in 3.7.3, that says: If $T^k=T$ for some integer $k\ge 2$ and if $w(T)\le 1$, then $T$ is the direct sum of a unitary ...
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some important proofs about adjoint operators [duplicate]

I was told that the formal adjoint of the gradient is the negative divergence. Let $A : H\to H$ be a bounded, linear operator, The adjoint of $A$, i.e. $A^*: H\to H$ satisfies \begin{equation*} ...
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Square root of differentiation

Let $T=d/dx$ be the differentiation operator on vector space $V=C^{\infty}(\mathbb{R})$, the space of real (complex) valued smooth maps on real line. To what extent, all subvector space ...
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A simple $C_{0}$-semigroup question.

Let $u:[0,t_{e}]\to\mathcal{D}(A)$ satisfy $$\begin{cases} \frac{du}{dt}=Au & 0\le t \le t_{e} \\ u(0)=x \end{cases}$$ I want to prove that necessarily $u(t)=T(t)x$. So it's clear to see that ...
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compare norms on $\mathcal{B}(H)$

Given a Hilbert space $H$ and $a$ be a real numbers $\geq‎‎‎ 1$ , let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T ...
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Show that the spectrum of an operator on $\ell^2(\mathbb{N})$ is $\{0\}$.

The problem I have comes from Walter Rudin's Functional Analysis, chapter 10 exercise 19. The exercise begins with the following: Let $S_R$ be the right shift operator, acting on ...
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25 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} + ...
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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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38 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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42 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 = ...
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96 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 ...
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32 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 ...