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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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 ...
2
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 ...
2
votes
1answer
45 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. ...
2
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1answer
51 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
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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 ...
5
votes
1answer
135 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 ...
10
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1answer
222 views
In a C*-algebra, put $a^*a \sim aa^*$. Transitivity fails?
Idle curiosity drove me to wonder about the following question. Let $A$ be a C*-algebra. Define a binary relation $\sim$ on the cone $A^{\geq 0}$ of positive elements by putting $x \sim y$ whenever ...
2
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1answer
38 views
Strong convergence of multiplication operator
I am looking for a necessary and sufficient condition for a sequence of multiplication operators $T^{(k)}$ to converge to zero strongly. (i.e. $\forall x \in \mathcal{H} \quad ||T^{(k)}x - 0|| \to 0$ ...
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2answers
70 views
How to prove that two non-zero linear functionals defined on the same vector space and having the same null-space are proportional?
Let $f$ and $g$ be two non-zero linear functionals defined on a vector space $X$ such that the null-space of $f$ is equal to that of $g$. How to prove that $f$ and $g$ are proportional (i.e. one is a ...
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3answers
161 views
How to find the norm of this bounded linear functional?
Let $C^\prime[a,b]$ denote the normed space of all continuously differentiable real (or complex) valued functions defined on the closed, bounded interval $[a,b]$ in $\mathbf{R}$ with the norm defined ...
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0answers
111 views
show that the function satisfies condition of the lemma
Let $(f_n)_{n\geq 0}$ be an eigenfunctions of the compact integral operator
$F$, defined on $L^2([-1,1])$ by
$$
F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here ...
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1answer
64 views
Compact resolvent
Given that the operator
$$
Hf(x) = -xf''(x) + (x - 1)f'(x)
$$
on the Hilbert space $L^2([0,\infty),e^{-x}dx)$ possesses, for each $n \in \mathbb{N}$, an eigenvalue $\lambda_n = n$ with eigenvector ...
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votes
1answer
167 views
Adjoint operator
This is about, a question I answered. Now there is an additional question that I cannot answer and do not want to spend any more time on. I feel like the question will not get any attention, as I ...
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1answer
56 views
Show that operator is continuous and selfadjoint (or not)
In this thread
Show compactness/ noncompactness of an operator by approximation
I came to the conclusion that the operator
$$
T\colon\ell^2\to\ell^2, ...
1
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1answer
98 views
What is the norm of this bounded linear functional?
Let $a$, $b$ be two arbitrary but fixed real numbers such that $a < b$, let $C[a,b]$ denote the normed space of all continuous real (or complex) valued functions defined on $[a,b]$ with the maximum ...
3
votes
1answer
51 views
Show compactness/ noncompactness of an operator by approximation
I have to show whether the following operator is compact or not:
$$
T\colon\ell^2\to\ell^2: (x_n)_{n\in\mathbb{N}}\mapsto\left(\frac{x_n+x_{n+1}}{2}\right)
$$
My idea was to approximate $T$ by ...
2
votes
2answers
84 views
Norm of differentiation operator $Tf(t)=f^{'}$..
Consider $T:C^1[0, 1]\rightarrow C[0, 1]$ given by $Tf=f'$ where $$\|f\|_{C^1}=\|f\|_\infty+\|f'\|_\infty$$ and $\|f\|_\infty=\sup_{x\in [0, 1]}|f(x)|$. How to prove $\|T\|=1$? The inequality ...
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2answers
43 views
Determine operator norm and show compactness
Consider
$$
T\colon\ell^1\to\ell^1, (s_n)\mapsto\left(\frac{s_{n+1}}{n}\right).
$$
Calculate the norm of $T$ and show that $T$ is compact.
1.) Operator norm of $T$
What I have is the ...
1
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1answer
70 views
What are the range and the norm of this bounded linear operator?
Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
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1answer
50 views
1.4.5 Theorem of Murphy's book
See 1.4.5 Theorem of Murphy's book : I want to prove that if $u$ be compact operator on $X$ which is Banach space and $\lambda\in \mathbb{C}\setminus\{0\}$, then ...
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1answer
78 views
How to find the range and inverse of this linear operator?
Given $T \colon C[0,1] \to C[0,1]$ defined by $$Tx(t):= \int_0^t x(r) dr$$ for each $t\in [0,1]$, where $C[0,1]$ is the normed space of continuous real-valued (or complex-valued) functions defined on ...
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1answer
35 views
A norm approximation for almost orthogonal operators
Let $H$ be a separable Hilbert space. Let $a,b: H\to H$ be bounded linear operators.
$a$ and $b$ are called orthogonal, if $a^*b=ab^*=0$. It is easy to see that this means that the support and image ...
3
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1answer
117 views
Normal operator + only real eigenvalues implies self-adjoint operator?
Let say we are in a complex vector space, is there an example of a normal operator with only real eigenvalues(or without eigenvalues) that is not a self-adjoint operator? Cause of the spectral theorem ...
2
votes
2answers
71 views
Show that operator is continuous
Show that
$$
V\colon H^{1,2}(\mathbb{R},\mathbb{R})\to\mathbb{R}
$$
is continuous, where
$$
V(u)=\int\limits_{-\infty}^{\infty}\left(\frac{1}{2}(\partial_x ...
1
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1answer
40 views
Adjoint series representation?
I am aware that for a normal square matrix $M\in\Bbb C^{n\times n}$, there exists a polynomial $P$ so that $P(M)=M^*$ What if I have a normal bounded operator $T\in\mathscr L(X)$ where $X$ is a ...
3
votes
0answers
67 views
invertible operator Sobolev space
Let $T:H_0^2(D)\rightarrow H_0^2(D)$ a linear bounded operator defined via the sesqulinear form $(Tu,v)=a(u,v):=\int_{D}\Delta u\Delta v dx$. I have shown that T is coercive and self-adjoint. How can ...
1
vote
1answer
53 views
Show that operator is normal and determine its Singular Value Decomposition
could anybody please help me with the following task?
Consider the operator
$$
Af(x):=\int\limits_{-\pi}^{\pi}\sin(x-y)f(y)\, dy, x\in [-\pi,\pi], f\in L_2(-\pi,\pi).
$$
Show that the operator ...
3
votes
1answer
72 views
Spectrum in Banach Algebra
Let $A$ be a unital Banach algebra and $a\in A$. Let $U$ be an open subset of $\mathbb C$ containing $\sigma (a)$. Prove that there is $\delta>0$ such that for every $b\in A$, if ...
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1answer
26 views
Is the adjoint of a quasinormal operator quasinormal as well?
I am trying to make sense of the various properties of operators on Hilbert spaces that generalise the notion of normality. It is known that for a (bounded) operator $A$ there are the following ...
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1answer
116 views
Trace of an operator
Suppose $\rho$ is a positve trace-class operator and $x$ is positve bounded operator on a Hilbert space $\mathcal{H}$, I am unable to prove that trace of $\rho x$ is positive,
where trace($x$):= $\sum ...
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votes
2answers
24 views
Projection of the third dual of a Banach space onto the first dual
Let $j_X:X\rightarrow X^{**}$ denote the canonical embedding.
I've read several articles where it is assumed that the reader is familiar with the idea that there is a norm one projection from ...
2
votes
3answers
99 views
Norm of bounded operator on a complex Hilbert space.
It is fairly easy to show that for a bounded linear operator $T$ on a Hilbert space $H$ $$||T||=\sup_{||x||=1,||y||=1}|\langle y, Tx \rangle |.$$
If $H$ is a complex Hilbert space, can you show that
...
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0answers
30 views
The deficiency indices of symmetric operators
Given any pair of nonnegetive integer $(a,b)$, can you find an (unbounded) symmetric operator $T$ with the deficiency indices $(a,b)$?
I guess the answer is yes, but how to do it?
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votes
1answer
45 views
Selfadjoint and continuous operator on a complex Hilbert space
Let $T\colon H\to H$ be a selfadjoint continuous operator on a complex Hilbert space. Show:
$$
\lVert (T\pm i\mbox{Id})x\rVert^2=\lVert Tx\rVert^2+\lVert x\rVert^2~\forall~x\in H.
$$
--
How can I ...
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votes
1answer
327 views
Rayleigh-Ritz Theorem
Let $U$ be an $n$-dimensional subspace of $L:=L_2([-1,1])$. Let $F$ be an acting on $L$, given at $f \in L$
$$
(Ff)(x):=\int_{-1}^1 \frac{\sin a(x-y)}{(x-y)}f(y) dy, \quad x \in [-1,1], \quad a>0.
...
1
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1answer
33 views
$P$ projector. prove that $\langle Px,x\rangle=\|Px\|^2.$
Let $X$ be a Hilbert space and $P \in B(X)$ a projector. Then for any $x\in X$:
$$\langle Px,x\rangle=\|Px\|^2.$$
My proof:
$$\|Px\|^{2}=\langle Px,Px\rangle=\langle P^{*}Px,x\rangle=\langle ...
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1answer
29 views
Show non-compactness of multiplication operator on $C[0,1]$
Show that the multiplication operator
$$
(Ax)(t):=(t+1)x(t)
$$
in the Banachspace $C[0,1]$ is not compact.
Again I am struggling with compactness, it is always difficult to me to decide ...
2
votes
0answers
70 views
Find the adjoint operator
I would like to find the adjoint operator in the Hilbertspace $L^2(0,\infty)$ of the operator
$$
(Ax)(t)=x(at), x\in L^2(0,\infty), a>0.
$$
My calculation is the following; I use the ...
0
votes
2answers
91 views
Example of a normal operator which has no eigenvalues
Is there a normal operator which has no eigenvalues?
If your answer is yes, give an example.
Thanks.
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1answer
54 views
Continuous, selfadjoint and compact?
Hell0 there!
I have to show whether the operator
$$
T\colon L^2(\mathbb{R})\to L^2(\mathbb{R}), f\mapsto\chi_{[0,1]}f
$$
is continuous, selfadjoint and compact.
I have problems to show the ...
2
votes
1answer
44 views
How can I prove that: $\|P_{1}x\|^2+\ldots+\|P_{n}x\|^2 \leq \|x\|^2$?
Let $X$ be a Hilbert space and $P_{j}\in B(X)$ a projector, for any $j=\overline{1,n}$. If $P=P_{1}+P_{2}+\ldots+P_{n}$ is projector, prove that for any $x \in X$:
$$\|P_{1}x\|^2+\ldots+\|P_{n}x\|^2 ...
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votes
2answers
45 views
$T:\ell^{2} \rightarrow \ell^{2}$ defined by $T(\{x_{n}\})=\{2^{-n}x_{n}\}$ is compact
Please help me to proof of problem :
Show that the operator $T:\ell^{2} \rightarrow \ell^{2}$ defined by $T(\{x_{n}\})=\{2^{-n}x_{n}\}$ is compact.
Tanks for your hint.
7
votes
3answers
202 views
Spectral radius inequality
Suppose $A,B \in M(n \times n, \mathbb{C})$ or $ A,B \in M(n \times n, \mathbb{R}) $. Under wich hypothesis can I state that:
$\rho(AB) \leq \rho(A)\rho(B)$ ?
2
votes
1answer
89 views
Gelfand's Formula. $r(T)=\lim_{n \to\infty}\sqrt[n]{\|T^{n}\|}$
Can you indicate me a material where I cand find the proof of Gelfand's Formula. I heard that there is a proof with polynomials.
Gelfand's Formula :
If $T \in B(X)$ then: $$r(T)=\lim_{n ...
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4answers
145 views
Operator theory: $T^2=T$, but $T^{*}\neq T$
Give an example of operator $T:\mathbb{R}^2\to\mathbb{R}^2$ with $T^2=T$, but $T^{*}\neq T$.
What could I consider ?
thanks :)
1
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2answers
51 views
Operator: $Tx =\sum_{n=2}^\infty x_{n-2} \frac{1}{2^{n-2}} e_{n}$. Which is $\rho(T)? $
Let $X$ be a separable Hilbert space and let $\{e_{k}: k\geq1\}$ be a orthonormat base in $X$. Let $T \in \mathcal{B}{(X)}$ defined by:
$$Te_{k}=2^{-k}e_{k+2}, k\in\mathbb{N}^{*}.$$
Find out the ...
2
votes
2answers
121 views
$\operatorname{Range}T$ is a closed subspace.
Let $X,Y$ two Banach spaces. If $T \in \mathcal{B}(X,Y)$ study if $\operatorname{Range}T$ is a closed subspaces.
How can I prove this fact ? What theorems can I use ?
thanks :)
1
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1answer
46 views
Operator question. $\sigma(T)\neq \varnothing.$
Let $X$ be a Banach real space and $T \in\mathcal{B}(X)$, where $T$ is an operator. Study if: $$\sigma(T)\neq \varnothing.$$
Can you help me please, thanks :)
2
votes
2answers
66 views
Compactness of multiplication operator on $C[0,1]$
Find a condition on function $a\in C[0,1]$ such that the operator $A:C[0,1]\rightarrow C[0,1]$ $$(Ax)(t) = a(t)x(t)$$ is compact? We are taking uniform norm on $C[0,1]$.
2
votes
0answers
128 views
Fredholm and Compact Operators
Let $X$ and $Y$ be Banach spaces and $T\in B(X,Y)$ be Fredholm. Then there is $S\in B(Y,X)$ such that $ST=I+K_{1}$ and $TS=I+K_{2}$ where $K_{1},K_{2}$ are compact operators.Proof: Since $T$ is ...
