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.

learn more… | top users | synonyms

0
votes
2answers
23 views

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: ...
1
vote
1answer
21 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 ...
2
votes
2answers
22 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 ...
3
votes
0answers
25 views

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 ...
2
votes
1answer
15 views

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 ...
1
vote
1answer
26 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 ...
1
vote
1answer
14 views

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 ...
0
votes
1answer
33 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 ...
1
vote
1answer
22 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 ...
0
votes
0answers
24 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 ...
4
votes
1answer
51 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 ...
0
votes
0answers
33 views

Differential Operator simplifying

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" by Elna Browning McBride In Sec 5" The extended form of the group generated by B and C " I did not understand ...
3
votes
0answers
41 views

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*} ...
5
votes
0answers
59 views

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 ...
3
votes
1answer
54 views

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 ...
1
vote
0answers
82 views
+50

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 ...
2
votes
1answer
46 views

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 ...
1
vote
0answers
17 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} + ...
2
votes
1answer
40 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 ...
0
votes
1answer
36 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 ...
2
votes
2answers
39 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 = ...
1
vote
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 ...
1
vote
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 ...
1
vote
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
votes
0answers
23 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
votes
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": ...
1
vote
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
votes
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) ...
1
vote
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 ...
-1
votes
0answers
23 views

A is an Hilbert operator, $(A+I)^{-1}$ is continuous with dense domain then A is essentially self-adjoint

We have no information about A, just that is an operator defined in $X\subset H$ where $H$ is a Hilbert space.
3
votes
1answer
50 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 ...
1
vote
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 ...
2
votes
1answer
69 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 ...
1
vote
1answer
47 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 $Y$ ...
1
vote
0answers
51 views

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 ...
1
vote
1answer
78 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 ...
0
votes
1answer
34 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 ...
0
votes
1answer
29 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: ...
0
votes
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 ...
4
votes
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 ...
2
votes
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 ...
2
votes
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\} ...
2
votes
1answer
26 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 ...
2
votes
2answers
69 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?
1
vote
1answer
78 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 ...
1
vote
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 ...
3
votes
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 ...
1
vote
2answers
26 views

Definition of associative algebra over a field

In the definition of an algebra over a field in the wiki entry , it states that an algebra over a field is a vector space equipped with a bilinear product. Question: Does anyone know how a bilinear ...
2
votes
0answers
28 views

Absolutely Continuous Spectrum and Norm of Resolvent

Problem. Let $H$ be a Hilbert space, and let $A:H\rightarrow H$ be a bounded, linear operator. Suppose $A$ has purely absolutely continuous spectrum and $\sigma_{ac}(A)=[0,1]$. Find the set of ...
0
votes
0answers
17 views

Is the following inequality true : $(||f(|A|)||)(||g(|A^*|)||) \geq ||A||$?

Please help me on this. Give an example of an operator $A\in B(H)$, such that $$(||f(|A|)||)(||g(|A^*|)||) < ||A||$$ where $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ ...