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
79 views
local convexity of $L_p$ spaces
wiki says The spaces $L_p([0, 1])$ for $0 < p < 1$ are equipped with the F-norm
they are not locally convex, since the only convex neighborhood of zero is the whole space
Why is this so? ...
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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 ...
2
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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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 ...
1
vote
1answer
71 views
norm equivalence
Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices ...
2
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$?
...
1
vote
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 ...
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
48 views
Injectivity of a certain operator
Consider a compact $K$ of the complex plane of interior void, $f$ a rational fraction leaving $K$ invariant (i.e. $f(K) \subset K$), and $\mu$ a borelian probability measure supported by $K$, and ...
0
votes
1answer
113 views
The $\alpha$-Potential-Operator (Definition and resolvent Equation)
during my studies I encountered the following Operator ($X_t$ is the standard Browniang Motion, $\alpha>0$ and $f$ is bounded function )
$U^{\alpha}f(x)=\mathbb{E}^x \int_0^{\infty} e^{-\alpha ...
2
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. ...
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 ...
5
votes
1answer
132 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 ...
2
votes
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
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 ...
2
votes
1answer
36 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. $\forall x \in \mathcal{H} \quad ||T^{(k)}x - 0|| \to 0$ ...
0
votes
3answers
160 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 ...
1
vote
2answers
69 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 ...
1
vote
3answers
346 views
An introductory textbook on functional analysis and operator theory
I would like to ask for some recommendation of introductory texts on functional analysis. I am not a professional mathematician and I am totally new to the subject. However, I found out that some ...
4
votes
0answers
183 views
Sum of operator and adjoint is self-adjoint
In abstract Hodge theory there is the following lemma:
Let $H$ be a Hilbert space and $A \in \mathcal{C}(H)$ a densely defined, closed operator (so possibly unbounded) and $A^*$ its adjoint operator. ...
1
vote
1answer
61 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 ...
0
votes
1answer
163 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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votes
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
$$
T\colon\ell^2\to\ell^2, ...
0
votes
1answer
50 views
1.4.5 Theorem of Murphy's book
See 1.4.5 Theorem of Murphy's book : I want to prove that if $u$ be compact operator on $X$ which is Banach space and $\lambda\in \mathbb{C}\setminus\{0\}$, then ...
1
vote
1answer
93 views
What is the norm of this bounded linear functional?
Let $a$, $b$ be two arbitrary but fixed real numbers such that $a < b$, let $C[a,b]$ denote the normed space of all continuous real (or complex) valued functions defined on $[a,b]$ with the maximum ...
3
votes
1answer
50 views
Show compactness/ noncompactness of an operator by approximation
I have to show whether the following operator is compact or not:
$$
T\colon\ell^2\to\ell^2: (x_n)_{n\in\mathbb{N}}\mapsto\left(\frac{x_n+x_{n+1}}{2}\right)
$$
My idea was to approximate $T$ by ...
2
votes
2answers
74 views
Norm of differentiation operator $Tf(t)=f^{'}$..
Consider $T:C^1[0, 1]\rightarrow C[0, 1]$ given by $Tf=f'$ where $$\|f\|_{C^1}=\|f\|_\infty+\|f'\|_\infty$$ and $\|f\|_\infty=\sup_{x\in [0, 1]}|f(x)|$. How to prove $\|T\|=1$? The inequality ...
1
vote
1answer
70 views
What are the range and the norm of this bounded linear operator?
Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
1
vote
2answers
43 views
Determine operator norm and show compactness
Consider
$$
T\colon\ell^1\to\ell^1, (s_n)\mapsto\left(\frac{s_{n+1}}{n}\right).
$$
Calculate the norm of $T$ and show that $T$ is compact.
1.) Operator norm of $T$
What I have is the ...
2
votes
2answers
71 views
Show that operator is continuous
Show that
$$
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 ...
3
votes
1answer
323 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 $f \in L$
$$
(Ff)(x):=\int_{-1}^1 \frac{\sin a(x-y)}{(x-y)}f(y) dy, \quad x \in [-1,1], \quad a>0.
...
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1answer
78 views
How to find the range and inverse of this linear operator?
Given $T \colon C[0,1] \to C[0,1]$ defined by $$Tx(t):= \int_0^t x(r) dr$$ for each $t\in [0,1]$, where $C[0,1]$ is the normed space of continuous real-valued (or complex-valued) functions defined on ...
2
votes
1answer
32 views
A norm approximation for almost orthogonal operators
Let $H$ be a separable Hilbert space. Let $a,b: H\to H$ be bounded linear operators.
$a$ and $b$ are called orthogonal, if $a^*b=ab^*=0$. It is easy to see that this means that the support and image ...
3
votes
1answer
113 views
Normal operator + only real eigenvalues implies self-adjoint operator?
Let say we are in a complex vector space, is there an example of a normal operator with only real eigenvalues(or without eigenvalues) that is not a self-adjoint operator? Cause of the spectral theorem ...
1
vote
1answer
40 views
Adjoint series representation?
I am aware that for a normal square matrix $M\in\Bbb C^{n\times n}$, there exists a polynomial $P$ so that $P(M)=M^*$ What if I have a normal bounded operator $T\in\mathscr L(X)$ where $X$ is a ...
2
votes
0answers
45 views
Show compactness of an evolution operator
Consider the heat equation
$$
u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$
with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$
and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$.
1.) ...
3
votes
0answers
67 views
invertible operator Sobolev space
Let $T:H_0^2(D)\rightarrow H_0^2(D)$ a linear bounded operator defined via the sesqulinear form $(Tu,v)=a(u,v):=\int_{D}\Delta u\Delta v dx$. I have shown that T is coercive and self-adjoint. How can ...
2
votes
2answers
120 views
$\operatorname{Range}T$ is a closed subspace.
Let $X,Y$ two Banach spaces. If $T \in \mathcal{B}(X,Y)$ study if $\operatorname{Range}T$ is a closed subspaces.
How can I prove this fact ? What theorems can I use ?
thanks :)
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vote
1answer
53 views
Show that operator is normal and determine its Singular Value Decomposition
could anybody please help me with the following task?
Consider the operator
$$
Af(x):=\int\limits_{-\pi}^{\pi}\sin(x-y)f(y)\, dy, x\in [-\pi,\pi], f\in L_2(-\pi,\pi).
$$
Show that the operator ...
3
votes
1answer
72 views
Spectrum in Banach Algebra
Let $A$ be a unital Banach algebra and $a\in A$. Let $U$ be an open subset of $\mathbb C$ containing $\sigma (a)$. Prove that there is $\delta>0$ such that for every $b\in A$, if ...
2
votes
1answer
26 views
Is the adjoint of a quasinormal operator quasinormal as well?
I am trying to make sense of the various properties of operators on Hilbert spaces that generalise the notion of normality. It is known that for a (bounded) operator $A$ there are the following ...
5
votes
1answer
115 views
Trace of an operator
Suppose $\rho$ is a positve trace-class operator and $x$ is positve bounded operator on a Hilbert space $\mathcal{H}$, I am unable to prove that trace of $\rho x$ is positive,
where trace($x$):= $\sum ...
2
votes
2answers
24 views
Projection of the third dual of a Banach space onto the first dual
Let $j_X:X\rightarrow X^{**}$ denote the canonical embedding.
I've read several articles where it is assumed that the reader is familiar with the idea that there is a norm one projection from ...
0
votes
1answer
142 views
There are compact operators that are not norm-limits of finite-rank operators
Given an example of a Banach space for which There are compact operators that are not norm-limits of finite-rank operators.
Tanks for answer
2
votes
3answers
96 views
Norm of bounded operator on a complex Hilbert space.
It is fairly easy to show that for a bounded linear operator $T$ on a Hilbert space $H$ $$||T||=\sup_{||x||=1,||y||=1}|\langle y, Tx \rangle |.$$
If $H$ is a complex Hilbert space, can you show that
...
4
votes
1answer
54 views
Continuous, selfadjoint and compact?
Hell0 there!
I have to show whether the operator
$$
T\colon L^2(\mathbb{R})\to L^2(\mathbb{R}), f\mapsto\chi_{[0,1]}f
$$
is continuous, selfadjoint and compact.
I have problems to show the ...
1
vote
0answers
30 views
The deficiency indices of symmetric operators
Given any pair of nonnegetive integer $(a,b)$, can you find an (unbounded) symmetric operator $T$ with the deficiency indices $(a,b)$?
I guess the answer is yes, but how to do it?
4
votes
1answer
44 views
Selfadjoint and continuous operator on a complex Hilbert space
Let $T\colon H\to H$ be a selfadjoint continuous operator on a complex Hilbert space. Show:
$$
\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
87 views
If a,b are unitary equivalent,Dose $\sigma(a)=\sigma(b)$ is true?
Let A is an unital algebra and $ Ad~u:A\rightarrow A~,~a\mapsto~uau^{*}$ and u is unitary element of A($uu^{*}=u^{*}u=1$), if $b=uau^{*}$ (a,b are ...
