Tagged Questions
4
votes
1answer
42 views
A problem on bounded invertible linear operator in Banach space
Let $X$ be a Banach space. Let $T : X \to X$ be a invertible
linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all
$k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x ...
1
vote
0answers
55 views
Proving properties of exponential map on a Banach algebra
$$\exp(a) := \sum\frac {a^k}{k!}$$
Can you help me prove that:
$\exp$ is well defined (i.e. converges for all $a$ in $A$)
$\exp$ is continuous
$\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
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. ...
0
votes
1answer
33 views
$\sigma(a)=\sigma(b)$, if $a,b$ are unitarily equivalent
Let $A$ be a *-algebra and $a,b$ are unitarily equivalent in $A$ ( i.e. there exists a unitary $u$ of $A$ s.t $b=uau^{*}$ ).
I want to prove that ...
0
votes
0answers
39 views
$\widehat{a}: \Omega(A)\rightarrow \mathbb{C}~,~\tau \mapsto \tau(A) $
Suppose that $A$ is abelian Banach algebra for which the space $\Omega(A)$ is non-empty. If $a \in A$, we define the function $\widehat{a}$ by $$\widehat{a}: \Omega(A)\rightarrow ...
1
vote
1answer
33 views
Gelfand representation Theorem
In proof of "Gelfand representation Theorem" (see 1.3.6 Theorem of Murphy's book ), I am understanding that why the map $$ A \rightarrow C_{0}(\Omega(A))~ , ~a\rightarrow ...
1
vote
1answer
107 views
If $X$ is an infinite-dimensional Banach space and $u\in B(X)$ ,then $\bigcap_{v\in K(X)}\sigma(u+v) =\cdots$
If $X$ is an infinite-dimensional Banach space and $u\in B(X)$,why the following equality is true?
$$\bigcap_{v\in K(X)}\sigma(u+v) =\sigma(u) \setminus \lambda \in\mathbb{C}\mid u - ...
-1
votes
1answer
76 views
The inclusion relation $\sigma(ab) \subseteq \sigma(a)\sigma(b)$ is not true for all Banach algebras
Let $A$ be a unital abelian Banach algebra. Give me an example that two following inclusion relations is not true for all Banach algebras
$$\sigma(a+b) \subseteq ...
1
vote
0answers
35 views
The hermitian element $h=\sum_{n=1}^\infty \frac{p_{n}}{3^{n}}$ generates $C_{0}(\Omega)$
Please help me to solve the following problem :
Let $\Omega$ be a locally compact Hausdorff space, and suppose that the $C^{*}$-algebra $C_{0}(\Omega)$ is generated by a sequence of projections ...
1
vote
1answer
35 views
A Linear map $u : X \longrightarrow Y$ is not bounded below iff there is …
Do you help me to:
checking that a linear map $u : X \longrightarrow Y$ between Banach spaces is not bounded below if and only if there is a sequence of unit vector ...
2
votes
2answers
69 views
If $A$ contains an idempotent $e$ ($e\neq 0,1$) , then $\Omega(A)$ is disconnected
If $A$ be a unital abelian Banach algebra and contains an idempotent $e$ (that is $e=e^{2}$) other than $0$ and $1$ , then help me to show that $\Omega(A)$ is ...
-3
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 ...
4
votes
0answers
84 views
invariant subspace of a Hardy space
Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...
4
votes
1answer
129 views
Fourier transform as a Gelfand transform
One question came to my mind while looking at the proof of Gelfand-Naimark theorem. Is Fourier transform a kind of Gelfand transform? Are there any other well-known transforms which are so?
4
votes
1answer
148 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 ...
2
votes
0answers
26 views
Initial topology of the spectrum mapping $\sigma$
Let $\mathcal{A}$ be a Banach algebra, the map $\sigma$ maps each element $a\in\mathcal{A}$ to its spectrum $\sigma(a)$, which is a compact subset of $\mathbb{C}$.
The collection of compact subsets ...
2
votes
2answers
55 views
Multiplication operators
Consider a commutative Banach algebra $A$ and the Banach algebra of bounded operators $B(A)$ on $A$. Associate to each $a\in A$ the multiplication operator $T_ax =ax$ ($x\in A$). Is always the mapping ...
4
votes
1answer
112 views
Schwarz inequality for unital completely positive maps
I came across the following form of Schwarz inequality for completely positive maps in Arveson's paper:
Let $\delta:\mathcal{A}\to\mathcal{B}$ be a unital completely positive linear map between ...
5
votes
2answers
137 views
Why are compact operators 'small'?
I have been hearing different people saying this in different contexts for quite some time but I still don't quite get it.
I know that compact operators map bounded sets to totally bounded ones, that ...
2
votes
0answers
45 views
Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?
Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation:
Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
2
votes
2answers
80 views
What does $(B+I)/I\sim B/(B\cap I)$ tell us?
Let $A$ be a $C^*$-algebra in which $B$ is a $C^*$-subalgebra and $I$ is a closed ideal. In several books on $C^*$-algebras I have encountered the following:
$(B+I)/I$ is $*$-isomorphic to ...
1
vote
1answer
79 views
Every ideal has an approximate identity?
Averson's 1970 paper on extensions of $C^*$-algebras seems to assume that every ideal has an approximate identity.
However, I am a little bit suspicious here, since he does not assume the closeness ...
0
votes
2answers
127 views
Linear functionals can be decomposed as linear combinations of positive ones?
I am reading Arveson's Notes on Extensions of $C^*$-algebras. In proving theorem 1, he needs to establish some results concerning bounded linear functionals. However, he said it suffices to prove for ...
3
votes
2answers
86 views
If $a\ge 0$ and $b\ge 0$, then $\sigma(ab)\subset\mathbb{R}^+$.
This is an exercise in Murphy's book:
Let $A$ be a unital $C^*$-algebra and $a,b$ are positive elements in $A$. Then $\sigma(ab)\subset\mathbb{R}^+$.
The problem would be trivial if the algebra ...
1
vote
1answer
61 views
References on Algebraic Operators
Let $\mathcal{H}$ be a Hilbert space and $d$ is an inner derivation on $\mathcal{L}(\mathcal{H})$. An operator $T\in\mathcal{L}(\mathcal{H})$ is algebraic if $p(T)=0$ for some polynomial $p$.
In ...
4
votes
1answer
50 views
Unital nonabelian banach algebra where the only closed ideals are $\{0\}$ and $A$
This is a problem in exercise one of Murphy's book
Find an example of a nonabelian unital Banach algebra $A$, where the only closed ideals are $\{0\}$ and $A$.
But does such an algebra exist at ...
0
votes
0answers
51 views
When is the orbit of a vector a minimal sequence? When does an operator have a minimal orbit vector?
Let $X$ be a banach space. We say a sequence $(x_n)$ is minimal if for each $k$, $x_k\notin [x_n]_{n\neq k}$, where $[x,y,z,\cdots]$ is the closed linear span of the vectors.
For ...
1
vote
1answer
150 views
The spectra of weighted shifts
Since weighted shifts are like the model-operators in operator theory and people have been studying them for so long, I think there should be quite a large literature on the spectra of such operators. ...
4
votes
1answer
130 views
A subset of $\bar{S}\backslash S$ contains an open ball in $\bar{S}$? (operator theory)
E and S are subsets of a metric space. $E$ is a subset of $\bar{S}\backslash S$. Then $\overline{E}\subset(\overline{S}\backslash S^{o})$, but I wonder whether there is some condition that guarantees ...
3
votes
1answer
211 views
Reflexive Banach algebras?
I have been reading Gelfand theory for a while and it just occurs to me that the whole theory is an analogy to what we did for Banach spaces.
For a Banach space $X$, we investigate its dual $X'$ ...
8
votes
1answer
432 views
Closure of the invertible operators on a Banach space
Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ ...
6
votes
1answer
308 views
Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $
Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $.
Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
9
votes
1answer
305 views
How to prove Halmos’s Inequality
How to prove Halmos’s Inequality?
If $A$ and $B$ are bounded linear operators on a Hilbert space
such that $A$, or $B$, commutes with $AB-BA$ then $$\|I-(AB- BA)\|\ge 1.$$
I found it from ...
4
votes
2answers
285 views
If $(I-T)^{-1}$ exists, can it always be written in a series representation?
If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm < $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$.
Thinking in terms of a ...
