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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21 views
Using “adjunction” to refer to the act of taking adjoints of operators
I have an especially flabby terminology question.
How acceptable is it, in your opinion, to use the word "adjunction" to refer to the process of taking adjoints of operators on a Hilbert space?
...
3
votes
1answer
127 views
Operators bounded below
Can one give me an easy example of an operator $T$ on a Banach space which is injective and has closed range and such that $\|T^2\|\neq \|T\|^2$?
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1answer
41 views
Operator defined via a sequence of weights
Let the linear operator $T:l^2\rightarrow l^2$ be defined by $y=Tx$ where $x=\{\xi_j\}$, $y=\{\eta_j\}$, and $\eta_j = \alpha_j \xi_j$, where $\{\alpha_j\}$ is a dense sequence in $[0,1]$. Does ...
5
votes
1answer
159 views
Every Hilbert-Schmidt is an integral operator?
Let $(X,\mu)$ be a $\sigma$-finite measure space. If $K\in\mathcal{L}^2(X\times X,\mu\times\mu)$ then the map $A_K:\mathcal{L}^2(X,\mu)\to\mathcal{L}^2(X,\mu)$ defined by\begin{equation}
...
8
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0answers
192 views
Limit of sequence of growing matrices
Let
$$
H=\left(\begin{array}{cccc}
0 & 1/2 & 0 & 1/2 \\
1/2 & 0 & 1/2 & 0 \\
1/2 & 0 & 0 & 1/2\\
0 & 1/2 & 1/2 & 0
\end{array}\right),
$$
...
3
votes
1answer
107 views
cyclic vector exists for symmetric operator iff there no repeated eigenvalues
Considering a symmetric operator $A$ acting on a finite dimensional Hilbert space $H$, we say $x\in H$ is a cyclic vector for $A$ if the set of finite linear combinations of $\{A^n x:n=0,1,2,...\}$ is ...
2
votes
3answers
138 views
Compact integral and multiplication operator in Banach spaces
Let $ A\colon C[0,1] \to C[0,1] $
$$ A(x)(t) = f(t)x(t) + \int_0^t x(s)ds,\quad f \in C[0,1]: f(1) \neq 0, \forall t \in [0,1] $$
Is $A$ a compact operator or not?
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votes
2answers
111 views
Operators with eigenvalue $\{0,1\}$ that is not projection
Show that there are linear operators T on the Hilbert space H what are not orthogonal projections, but their spectrum consists of the eigenvalues $\{0,1\}.$
I can not come up with an counterexample, ...
1
vote
1answer
64 views
Power series of bounded linear maps
Given a Banach space $X$ and a bounded linear map $T:X\rightarrow X$ we define $$e^T = I + \sum_{n\geq1}\frac{T^n}{n!}$$
Show that if $e^T$ is compact then dim $X<\infty$.
I have showed before ...
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0answers
63 views
Spectral raius for linear compact maps
Prove or disprove the following assertions for a linear map $C$ from a Banach space $X$ into itself:
a) If C is compact then its spectral radius equals the maximum of the absolute value of $C$
Im ...
3
votes
1answer
199 views
Operators on $C([0,1])$ that is compact or not.
For $f\in C([0,1])$ set
$$Hf(x) = \frac{1}{x}\int_0^x f(t)dt.$$
a) Show that $H$ is a bounded operator from $C([0,1])$ into itself which is not compact.
b) From a) it follows that $H$ induces a ...
3
votes
1answer
89 views
Compute spectral/projection-valued measures explicitly?
Spectral/projection-valued measures have very handy applications theoretically, but I got stuck when asked to compute explicitly certain projection-valued measures. Let's focus on the following:
...
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votes
1answer
42 views
Approximating bounded operators in Hilbert space
Let $H$ be a separable Hilbert space, show that every bounded operator from H to itself can be approximated in the strong operator topology by a sequence of finite rank operators.
I know we can find ...
2
votes
1answer
59 views
Graph of symmetric linear map is closed
A homework problem:
Let $H$ be a Hilbert space.
Let $T:H\rightarrow H$ be a symmetric linear map ($\langle Tx,y\rangle=\langle x,Ty\rangle$).
Show that $S$ is bounded.
My attempt: I'd ...
1
vote
1answer
52 views
Characterize compact sequences for a linear map.
Given a bounded sequence $\pi = (\lambda_n)$ in $\mathbb{C}$ consider the continuous linear map $M_\pi:\ell^2\rightarrow \ell^2$ defined by $$M_\pi(x_n) = (\pi_nx_n)$$
a) determine the spectrum.
b) ...
1
vote
1answer
86 views
The Principle of Condensation of Singularities
Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j ...
1
vote
1answer
94 views
The convergence of the adjoint operator
If a sequence of operator $A_n$ converges in norm to $A$, i.e. $\lim \lVert A_n-A\rVert=0$)where $A_n$ and $A\in B(H)$ ($H$ is the Hilbert space). Is it true that $A_n^*$ converges in norm to $A^*$?
1
vote
1answer
113 views
Normal operators in Hilbert spaces
Let $H$ be a separable Hilbert space and let $T:H\to H$ be a continues linear map such that there exists an orthonormal basis of $H$ that consists of the eigenvectors of $T$. Show that $T$ is normal. ...
2
votes
1answer
100 views
Determine the operator T in a Hilbert space
Let $H$ be a Hilbert space and let $\{e_n, n \geq 1\}$ be an orthonormal basis for $H$.
a) Determine the operator $T\in B(H)$ that satisfies
$$ Te_1 = 0,\; Te_n = \frac{1}{n}e_{n-1}, n ...
3
votes
2answers
167 views
Projection operator in Hilbert spaces
Let T be a bounded operator on the Hilbert space H with the property that $T^*(T-I)= 0$. Show that T is an orthogonal projection.
Im not really sure how to show that an operator is an orthogonal ...
2
votes
0answers
53 views
Convergence of a series in the $C^\infty$ topology
I am struggeling to understand the following argument given by F.Treves in the book "Introduction to Pseudodifferential Operators".
The motivating problem for this is to find an approximate kernel ...
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votes
1answer
76 views
Show that $(x_n)$ is in $\ell^2$
Let $x = (x_n)$ be a sequence of complex numbers with the property that for every $y = (y_n) \in \ell^2$ we have that the sequence $(S_N(y))_{N\geq1}$ with
$$S_N(y) =\sum_{n=1}^N x_ny_n $$ converges. ...
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votes
1answer
139 views
If $A_n$ is a sequence of positive bounded linear operators converging in norm to $A$ on a Hilbert Space, show $\sqrt{A_n}\to\sqrt{A}$ in norm.
If $A_n$ is a sequence of positive bounded linear operators converging in norm to $A$ on a Hilbert Space, show $\sqrt{A_n}\to\sqrt{A}$ in norm. I can show that $A$ would be positive and thus have a ...
3
votes
2answers
79 views
If a map $C:X\rightarrow U$ maps every weakly convergent sequence into strongly convergent
A Linear map between Banach spaces $C:X\rightarrow U$ is compact if it maps if the closure of the image of the unit ball is precompact in U.
If a map $C:X\rightarrow U$ maps every weakly convergent ...
3
votes
1answer
71 views
Does $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for bounded operators on Hilbert space?
If $A$ is a bounded linear operator on a Hilbert space $H$ is it true that $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for all $x\in H$? If not, can we at least establish inequality in one ...
2
votes
1answer
59 views
Unbounded sets in infinite-dimensional normed spaces.
Let $X$ be an infinte-dimensional normed space. Let $\ell_1,\ldots, \ell_n$ be continuous linear functionals on $X$ and consider the set
$$U = \{x\in X : |\ell_j(x)| < 1,\;\; 1\leq j \leq n\}.$$
...
2
votes
1answer
68 views
Collection of linear functions
Let $X$ be a Banach space. Let $\{Y_\alpha\}_\alpha$ be normed spaces.
Let $\{T_\alpha:X\rightarrow Y_\alpha\}_\alpha$ be an infinite collection of bounded linear functions.
Is there a way to create ...
2
votes
1answer
153 views
Transpose of Volterra operator
I want to find the transpose of the Volterra operator $$Vf(x) = \int_0^x f(t)dt, \;\; x\in(0,1)$$ acting in $V:L^2(0,1) \rightarrow V:L^2(0,1) $. The transpose is defined as $\textbf{M}':U'\rightarrow ...
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0answers
51 views
Finite Dimensional TVS
Let $E, F$ topological vector spaces, $E$ normable and $T: E \longrightarrow F$ linear, compact and surjective. Show that $\mbox{dim}(F)< \infty$.
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0answers
51 views
Compactness of multiplication operator [duplicate]
Possible Duplicate:
Compactness of Multiplication Operator on $L^2$
Let $u: \mathbb{R}\rightarrow \mathbb{C}$ be a bounded continuous function. Show that the multiplication operator $M_u$ ...
2
votes
1answer
101 views
Strong limit of compact operators
$X$ and $U$ are Banach spaces. A linear map $\textbf{C} : X \rightarrow U$ is called compact if the image $\textbf{C}B$ of the unit ball $B$ in X is precompact in $U$.
A subset S of a complete metric ...
4
votes
1answer
149 views
$\sigma(x)$ has no hole in the algebra of polynomials
Let $A$ be a unital banach algebra generated by two elements $1$ and $x$. Then it seems $\sigma(x)$ cannot have holes. At least this is true in the case for disk algebras.
This amounts to prove that ...
1
vote
1answer
41 views
Transpose of the Shift operators
Let $X = \ell^2$ The operators $\textbf{L}$ and $\textbf{R}$ are defined as
$$\textbf{R}x = (0, a_0, a_1...) \;\; \textbf{L}x = (a_1, a_2, a_3...) $$
show that they are the transposes of one another ...
2
votes
2answers
104 views
Bounded functionals on Banach spaces.
Let $(X, \|.\|)$ be a Banach space such that
$X \subset C([0,1]) $
For every $r\in \mathbb{Q}\cap[0,1], f\rightarrow f(r)$ defines a bounded linear functional on $X$.
Prove that there exists a ...
1
vote
1answer
112 views
Bounded integral operators in Functional analysis
Let $K: [0,1] \times \mathbb{R}^n \rightarrow \mathbb{C}$ have the properties:
1. $K(x,.) \in L^2(\mathbb{R}^n)$ for all $x\in[0,1]$
2. for every $f\in L^2(\mathbb{R}^n)$ the function
$$ x\rightarrow ...
2
votes
0answers
95 views
(SOLVED) Adjoint of Frechet derivative (involving gradient operator)
I need some help with a problem (a homework/programming exercise) regarding the adjoint operator of the Frechet derivative of an operator.
I have the forward operator $ F(a) = L_a ^{-1}f $ where ...
3
votes
1answer
121 views
Functional analysis summary
Anyone knows a good summary containing the most important definitions and theorems about functional analysis.
1
vote
1answer
80 views
Continuous operator on $L^\infty$
$1<p<\infty$ and $k\in L^\infty([0,1]^2)$
$(Tf)(s)=\int_{0}^{1}k(s,t)f(t)dt$
I want to show that it is a continuous operator $T:L^p([0,1]->L^p([0,1])$
Proof: What I need to show is that ...
2
votes
1answer
91 views
Open mapping theorem and second category
This seems like a fundamental result but I can not solve it of find an solution:
Let $M:X\rightarrow U$ be a bounded linear map between Banach spaces. Show that if the
range of M is a set of second ...
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1answer
36 views
Reference about Fredholm determinants
I am searching for a reference book on Fredholm determinants. I am mainly interested in applications to probability theory, where cumulative distribution functions of limit laws are expressed in terms ...
2
votes
0answers
43 views
Closed range for maps between banach spaces? [duplicate]
Possible Duplicate:
Question about Fredholm operator
This seems to be a standard result but I cannot find the solution.
Let $M:X \rightarrow U$ be a bounded linear map between two Banach ...
2
votes
1answer
130 views
Show that the Volterra operator have dense range.
Let $V: C([0,1]) \rightarrow C([0,1])$ be defined by
$$ V f(x) = \int\limits_0^x f(t) dt.$$
Show that V has dense range and find the transpose of V.
V has dense range: Since the polynomials are dense ...
0
votes
1answer
27 views
Finding operator with specific properties
Let $H=(\mathbb R^2,(.,.))$ and $M=\{(x,0)|x\in\mathbb R\}, N=\{(x,x\tan(\theta)|x\in\mathbb R)$ with $\theta\in(0,\frac{\pi}{2})$.
Now I would like to find a $T_\theta\in B(H,H)$ with ...
2
votes
0answers
113 views
Must-read papers in Operator Theory
I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some ...
3
votes
2answers
116 views
Proof that certain operators are compact
I want to examine which of the following operators $T \colon C[0,1] \to C[0,1]$. are compact, by some I think I got the argument, but others I have no idea.
a) $Tx(t) = x(t^2)$
Guess it is ...
1
vote
1answer
130 views
Neumann series and spectral radius
I have a question about the convergence of the Neumann series:
Let $A$ be a matrix with spectral radius $\rho(A)<1$, i.e., all eigenvalues of $A$ are strictly less than $1$. Does that imply that ...
8
votes
0answers
177 views
Inverse of Toeplitz Matrix Property
Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form
$$\left[\begin{array}{llll}
a_0 & a_1 & \dots & a_n\\
a_1 & a_0 ...
1
vote
1answer
70 views
Inverse of trace class operator restricted to it's range
A paper I'm reading constructs the Cameron-Martin space in a way different than I'm used to, and in the process they gloss over a functional analysis result about the existence of an inverse. It ...
3
votes
3answers
159 views
Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional
Suppose that we have a Hilbert Space $H$ and $M$ is a closed subspace of $H$. Let $T\colon H\rightarrow M$ be the orthogonal projection onto $M$.
I have to show that $T$ is compact iff $M$ is finite ...
3
votes
1answer
309 views
Hilbert-Schmidt Operator
We have just covered Hilbert-Schmidt operators in class (which I missed) and I am having a hard time getting my head around them. I know the definition:
If $H$ is a Hilbert space and ...
