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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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 ...
3
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1answer
244 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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78 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
119 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 ...
3
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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
75 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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357 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
168 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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34 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 $\phi$ ...
2
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1answer
179 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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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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91 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
325 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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46 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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35 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
93 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
46 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
141 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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146 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
44 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 ...
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243 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
65 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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149 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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53 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 ...
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1answer
106 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
62 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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99 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
42 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
82 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
78 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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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
69 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?
3
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1answer
167 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
70 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 $\lim_n\|T_n\|=0$.
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164 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
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275 views

Are there non nilpotent operators with spectrum 0?

If a linear operator on a vector space V is nilpotent, then its spectrum is 0. Makes me wonder, are there also operators with spectrum 0 that are not nilpotent? Necessarily such an operator is not ...
2
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1answer
61 views

What does this phrase about the weak topology of bounded operators mean?

Can somenone remind me of the meaning of the following statement: the family of operator valued functions $A(\omega)$ converges to $A(\omega ')$ in the weak topology of bounded operators from ...
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1answer
60 views

problem related to tensor product on Hilbert spaces

Let $K$ and $H$ be Hilbert spaces. Let $\{e_i:i\in I\}$ be an orthogonal basis of $H$. Define $$ U_i:K\to K\overset{.}{\otimes} H: x\mapsto x\overset{.}{\otimes} e_i $$ Assume ...
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1answer
77 views

Convergence in norm operator

If I have an operator valued functions $A(z):H_1\to H_2$ such that the following limit $$\lim_{z\to z'}A(z)=A(z')$$ exists in the uniform topology of $B(H_1,H_2)$, that is $$\Vert ...
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1answer
225 views

norm equivalence

Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices ...
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636 views

Spectral radius of the Volterra operator

The Volterra operator acting on $L^2[0,1]$ is defined by $$A(f)(x)=\int_0^x f(t) dt$$ How can I calculate the spectral radius of $A$ using the spectral radius formula for bounded linear operators: ...
4
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1answer
171 views

Normal $T\in B(H)$ has a nontrivial invariant subspace

I am wondering if the following is true: Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
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1answer
141 views

The span of the orthorgonal projections is norm dense in $B(H)$

This is a question in my functional analysis book. How to use the spectral theorem to prove that the span of the orthogonal projections is norm dense in $B(H)$?
3
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1answer
124 views

Operator norm and spectrum

Let $L$ be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of $L$ in terms of the spectrum of the positive operator $L^{\dagger}L$? ...
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1answer
80 views

generalized functions & operators

I am dealing with a function $f(r) $that behaves like ~ $\frac{1}{r}$ when approaching zero. When I take the Laplacian of this guy and then integrate the result ([0,$\infty$]) I get some additional ...
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1answer
101 views

Self-adjoint operator and inner product

I am wondering whether there is a way to make sense of self adjointness of an operator on $C[0,1]$ without resorting to the inner product of $L^2[0,1]$. I am not referring to concrete alternative ...
2
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1answer
59 views

Spectrum in Hilbert space

Let $H$ be a Hilbert space and $T\in B(H)$ be normal. Want to show that if $\lambda$ is an isoloated point in $\sigma(T)$ then $\lambda$ is an eigenvalue of $T$.I have no idea how to do this one. ...
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1answer
95 views

Counterexample for a polar decomposition in von Neumann and $C^\ast$ algebras

For a von Neumann algebra, we have that partial isometry and positive operator of an operator in its polar decomposition belongs to the algebra, but in a $C^\ast$ algebra this may not be true. Can ...
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1answer
88 views

Is this gradient an isomorphism on its range?

Consider the gradient (in the weak sense) as an operator $\nabla \colon H^1(\Omega)/\mathbb{R} \to [L^2(\Omega)]^d$, where $\Omega \subset \mathbb R^d$ is a domain with a smooth boundary and ...
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444 views

Symmetric Square Root of Symmetric Invertible Matrix

I am trying to find out if for any symmetric (Not necessarily self-adjoint), invertible matrix $A$ over $\mathbb{C}$, there is a square root of the matrix that is also symmetric. I was able to figure ...