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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70 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 ...
3
votes
1answer
110 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
63 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
57 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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votes
1answer
41 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
votes
1answer
45 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) ...
4
votes
1answer
54 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 ...
4
votes
2answers
74 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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0answers
28 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
votes
1answer
48 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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38 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
63 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
votes
1answer
172 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
39 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
25 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
43 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
37 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
35 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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0answers
28 views
Strange equality of the operator E($Eu_n=u_{n+1}$)
The operator $E$ is defined as $Eu_n=u_{n+1}$.
I encountered a strange equality. when I tried out
Let $u_n$ represent a series such that
$$u_{n+2}=u_{n+1}+u_n. \tag{$\star$}$$
Or
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119 views
Prove that the integral operator is bounded
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
27 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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votes
1answer
120 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
49 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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82 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
35 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
53 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
44 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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votes
2answers
61 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
32 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
45 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
38 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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44 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
47 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
52 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
60 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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74 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 ...
4
votes
2answers
100 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 ...
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votes
1answer
51 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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votes
1answer
54 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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vote
1answer
42 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
$$\Vert ...
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1answer
71 views
norm equivalence
Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices ...
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127 views
Spectral radius of the Volterra operator
The Volterra operator acting on $L^2[0,1]$ is defined by $$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: ...
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1answer
69 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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votes
1answer
58 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
votes
1answer
66 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
42 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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votes
1answer
57 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 ...
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votes
1answer
44 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
50 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 ...
3
votes
1answer
54 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 ...
