Tagged Questions

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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1answer
59 views

A basic result about operators on Hilbert space.

I am studying following result. Let $H$ and $K$ be Hilbert spaces and an operator $A \in B(H, K)$, which has closed range. The spaces $H$ and $K$ have the following orthogonal decompositions: $H = ...
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0answers
29 views

The Square of the Laplace Transform

I have been looking at the Laplace transform $$\mathcal{L}f(s)=\int_0^{\infty}f(t)e^{-st}dt$$ and I'm trying to find The norm of $\mathcal{L}^2$ The nullspace of The norm of $\mathcal{L}^2$ So ...
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1answer
97 views

Strong operator convergence and adjoint operator

Let $H$ be a Hilbert space and $(T_n)_{n \in \mathbb{N}}$ be a sequence of bounded linear operators on $H$. The strong convergence of $T_n$ doesn't imply the strong convergence of $T_n^*$, i.e. ...
3
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1answer
179 views

Invertibility of compact operators

I'm a little confused about compact operators and whether or not they are invertible. Just hoping someone here can clear up my confusion: Let $T$ be a compact operator on a Banach space $X$. Since ...
3
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1answer
101 views

$K$ is a linear compact operator on Hilbert space $H$. Will the image of $I-K$ on every closed subspace of $H$ be also closed?

Just as the title. We know the image of $I-K$ is closed, but if we restrict $H$ to a closed subspace $V$, will $(I-K)(V)$ be a closed subspace of $H$? Any hint is appreciated.
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1answer
95 views

Proof of implication: $\varphi^*\text{ is bounded below}\implies\varphi\text{ is a quotient map}$

We say that a bounded operator $\varphi:X\to Y$ is $c$-topologically injective if $\Vert\varphi(x)\Vert\geq c\Vert x\Vert$ for all $x\in X$ $c$-topologically surjective if for all $y\in Y$ there ...
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2answers
175 views

Composition of pseudo-differential operators

Denote by $S^m$ the set of functions $p: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}$ such that $p \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ and $$\left| ...
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2answers
301 views

Square root is operator monotone

This is a fact I've used a lot, but how would one actually prove this statement? Paraphrased: given two positive operators $X, Y \geq 0$, how can you show that $X^2 \leq Y^2 \Rightarrow X \leq Y$ (or ...
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0answers
193 views

Inverse of Identity plus Volterra operator

consider the following operator or $L_2(0,1)$, $(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial. I am trying to construct the inverse of this ...
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1answer
95 views

does invertibility of product imply invertibility of each term of product?

Suppose $\mathcal{H}$ is a Hilbert space and the product $T_1T_2 \in B(\mathcal{H})$ is invertible. Does this imply that both $T_1, T_2$ are invertible ? I am unable to prove this since, unlike the ...
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0answers
185 views

Is the sum of two closed operators closed?

If A and B are closed linear operators from $X$ to $X$ ($X$ is a normed vector space and the domain of them is X), is $A+B$ a closed operator? I think it's not but I can't find a counterexample. In ...
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193 views

Proving properties of exponential map on a Banach algebra

$$\exp(a) := \sum\frac {a^k}{k!}$$ Can you help me prove that: $\exp$ is well defined (i.e. converges for all $a$ in $A$) $\exp$ is continuous $\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
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1answer
56 views

Matrix completion: supplementary questions

Continuation of the question here, what is going to happen if we change the some of the conditions. I write it as a quote from here and change the appropriate places which are underlined: I need ...
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1answer
65 views

Exponential of nth order derivative

When dealing with an exponential operator of the form $e^{\vec a \cdot \vec \nabla}f(\vec x)$, I understand how this simply shifts the argument of the function by $a⃗$ . My question is what happens ...
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1answer
199 views

hoffman-and-kunze [chapter 8.2][question 4]

The question: Let Y be a finite-dimensional inner product space and T a linear operator on Y. Show that the range of T* is the orthogonal complement of the null space of T Think i got one way: took ...
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1answer
282 views

Show $T$ is invertible if $T'$ is invertible where $T\in B(X)$, $T'\in B(X')$

Seems simple enough but I can't quite get it. $X$ is a complex Banach space, and $T\in B(X)$, $T'\in B(X')$ is its adjoint. Suppose $T'$ is invertible. How can we show that $T$ is invertible? I have ...
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1answer
249 views

Spectral theorem of compact operators in Hilbert space

I am reading the following theorem from my lecture notes (English translation of German text). But I don't understand exactly what is meant from this theorem and the proof. Theorem. Let $H$ ...
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2answers
80 views

Completeness of Operator space.

Assume $(X,\|\cdot\|_X),(Y,\|\cdot\|_Y)$ are normed spaces and $\dim X\geq 1$. The following holds: $Y$ complete $\iff$ $\mathscr L(X,Y)$ complete. The latter denotes the space of bounded operators ...
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1answer
127 views

Image of a set under a mapping

I need to show that the image of the closed unit ball in $\mathbb{C}$, under the polynomial mapping $p(x) = (1-x)^2$ is the cardioid: ${re^{i\theta} : 0 \leq \theta < 2π, 0 ≤ r ≤ 2 + 2 ...
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1answer
51 views

existence of invertible operator mapping one sequence pointwise to a 'nearby' sequence

Let $X$ be a Banach space and $(x_n)$, $(y_n)$, $(f_n)$ be bounded sequences in $X$, $X$, $X^*$ respectively such that $f_m(x_n)=\delta_{mn}$ $\forall m,n$ and $\epsilon=\Sigma\|x_n-y_n\|<\infty$. ...
3
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1answer
77 views

Spectrum of the Hill Operator $L(y)= -y''+ v(x) y $

Consider the eigenvalue equation for the Hill operator $$L(y)= -y''+ v(x) y = \lambda y, \quad x\in \mathbb{R},$$ where $v(x)$ is any potential and $\lambda$ is the spectral parameter. If $v(x) ...
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1answer
400 views

Closed unbounded operator with domain not closed

I am looking for an example for further understanding of the Closed Graph Theorem: Let $X,Y$ be Banach spaces and $T:X\to Y$ closed (i.e. the graph of $T$ is closed in $X\times Y$). Then if ...
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2answers
192 views

Show that linear Operator on $\ell^2$ is unbounded

Currently, I am preparing for a next semester course and trying to figure out some basic concepts in functional analysis. Let $T:\mathcal{D}(T)\to \ell^2$ be defined by ...
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35 views

A map that is $(n-1)$-positive but not $n$-positive

Let $\phi : M_n(\mathbb{C})\to M_m(\mathbb{C})$ be a linear map. $\phi$ is called $k$-positive if the map $\phi^{(k)} : M_{kn}(\mathbb{C}) \to M_{km}(\mathbb{C})$, defined by evaluating $\phi$ ...
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1answer
189 views

Closed range operators

Let $T$ be a linear operator between two normed spaces. I'm trying to show that an operator $T$ has closed range if and only if $\operatorname{im}(T) = (\ker{(T^*)})^{\perp}$. Is there a way to do it ...
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0answers
43 views

Operator Graph Question

Let $T$ be closable. I am trying to show $\Gamma(\overline{T}) \subseteq \overline{\Gamma(T)}$. I can already show the reverse inclusion. Any ideas?
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1answer
94 views

Is there any operator which its spectrum corresponding to a compact set?

we know that for each operator $T$ the spectrum $\sigma(T)$ is compact. Is the converse true I mean if we have a compact set $K\neq\emptyset$, is there any operator $T$ such that $\sigma(T)=K$? I am ...
5
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1answer
343 views

Bounded operator and Compactness problem

Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator. a) Let $x\in [a,b]$. Show that there is a ...
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0answers
49 views

Doubt about the spectrum of an operator

I consider the Laplacian operator $$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
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0answers
36 views

Finding Strictly Positive Elements [duplicate]

I need to find the set of strictly positive elements in the $C^*$-algebra $C_0(\Omega)$ where $\Omega$ is a locally compact Hausdorff space. Clearly, the set will be contained in $ \{ f \in ...
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2answers
115 views

example of positive but not completely positive operator

I was looking for some example of a positive operator which is not completely positive on a banach algebra. if I consider my banach algebra to be $\text{M}_n(\mathbb{C})$ of matrices over complex ...
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2answers
47 views

Structure of $L_1(G)$

I came across this while going through some basic examples of $C^*$ algebras. If I consider $G$ as the set of cube roots of unity, what will be $L_1(G)$? I mean what will be the structure of elements ...
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1answer
168 views

Norm of oblique projector and angle between subspaces

Take $V$ and $W$ closed subspaces of $H$ a Hilbert space with $V\oplus W=H$ (we'll assume this holds in the sequel, it may not be required everywhere but in the context of interest, it is always ...
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147 views

Prove that the integral operator is bounded [duplicate]

Prove that the following operator is bounded on $L^{2}(0, \infty)$: $Af(x)$ = $\frac{1}{\pi} \int_{0}^{\infty} \frac{f(y)}{x+y}dy$ with $||A|| \le 1$. Attempt at Solution It can be shown that: ...
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1answer
45 views

Property of sequence of eigenvalues of an operator.

For a positive (self adjoint) operator $A$ with eigenvalues $\lambda_k$, is it possible to have the case when neither $\lambda_k\to \infty$ or $sup_k \lambda_k<\infty$ for example if a subsequence ...
4
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1answer
251 views

Norms involving positive operators

Let's say we have $A \leq B$. Is it then true that $||Ax|| \leq ||Bx||$ (where $x, A, B$ all belong to the same finite-dimensional Hilbert space $H$)?
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1answer
66 views

Help with proving: If $X$ is a Hilbert $A$-$B$-module, then $ \| _A \langle x,x \rangle \| = \| \langle x,x \rangle _B \| $ for all $x\in X $.

Sorry, I posted a related question last week on here, but I'm still having trouble and this is a little different, I hope it's OK. Thank you! ( proof that this is an isometric map (on a $C^*$-module) ...
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0answers
152 views

When are two operators simultaneously diagonalizable?

I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously ...
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0answers
58 views

2 positive decomposable maps

A positive map $\phi:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ is said to be $k$-positive if the natural extension ...
2
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1answer
110 views

finding operator norm $T_N$

How do find the operator norm of , $T_N\colon c_0\to \Bbb R$ given by $T_N(y):=\sum_{j=1}^Nx_jy_j$ ,when $N$ is a integer . TIA
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1answer
67 views

Operators and Functions

What is the exact difference between operators and functions ( if there is any ) ? Can i say an operator is more general than a function as it turns functions into functions ( like the derivative ...
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2answers
102 views

System of equations wrt self-adjoint operators

$X = \left( \begin{matrix} 2&s\\ 8&2 \end{matrix} \right)$ and $Y = \left( \begin{matrix} 2&-1\\ 2&2 \end{matrix} \right)$ are two operators wrt the same orthonormal basis $B$ in a 2D ...
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1answer
44 views

Help proving operator inequality

Given $P \geq 0$, I need to show that $2Tr(P^{5/2}) \leq Tr(P^3) + Tr(P^2)$. It's trivial to show that the RHS is the trace of a positive operator, but I'm at a loss on how to actually prove this ...
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2answers
84 views

Multiplicative linear functional on algebra of limit of polynomials

Let $A$ be the space of all functions which are limit of polynomials over the unit ball $D$. Then $A$ is a commutative Banach algebra. Then how do I show that $A$ has no non zero multiplicative linear ...
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1answer
86 views

Linear functional on Banach algebra

Let $A$ be the space of all matrices of the form $\begin{pmatrix} a & b \\0 & a\end{pmatrix}$, $2\times2$ over complex field. Then the spectrum of any element of $A$ comes out to be $\{a\}$. I ...
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4answers
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Are all algebraic commutative operations always associative? [duplicate]

I know that there are many algebraic associative operations which are commutative and which are not commutative. for example multiplications of matrices as associative operation is not commutative. I ...
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0answers
95 views

Operator monotone functions

By definition, I know that a function $f$ is operator monotone if $A - B \geq 0 \Rightarrow f(A) - f(B) \geq 0$. For instance, we have $A^2 \leq B^2 \Rightarrow A \leq B$ because the root function is ...
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1answer
71 views

Operator inequalities: $0 \leq A \leq B \Rightarrow Tr(A^p) \leq Tr(B^p)$?

It is trivial to show that $0 \leq A \leq B \Rightarrow Tr(A^2) \leq Tr(B^2)$, but does this generally hold for all $p >$ 2 as well?
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1answer
184 views

How to prove the compactness of this Sobolev embedding?

I have a question on compactness of the following Sobolev embedding. Let $W^{1,p}([0,1],\mathbb{R}^n)$ be the Sobolev space of functions $u:[0,1]\rightarrow \mathbb{R}^n$ for $1<p<\infty$. How ...
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2answers
72 views

Operator norm converging to 0 for certain condition

Let $X$ be a finite-dimensional normed space and $T_n : X \to X$ a sequence of linear operators such that $\lim_nT_nx = 0$ for all $x$ in $X$. Prove that $\lim_n\|T_n\|=0$.