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Questions tagged [normal-operator]

For question concerning normal operators in Hilbert spaces.

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If $\lambda$ is an eigenvalue of a normal operator $A$, then $\left.A\right|_{{\mathcal N(\lambda-A)}^\perp}$ is well-defined and normal as well

Let $H$ be a $\mathbb R$-Hilbert space, $A$ be a normal$^1$ linear operator on $H$ and $\lambda\in\mathbb R$ be an eigenvalue of $A$. Are we able to show that ${\mathcal N(\lambda-A)}^\perp\...
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117 views

When a normal operator is also a self adjoint operator?

Let $T$ be a normal operator on a complex inner product space. Then $T$ is a self adjoint operator if and only if 1) $T$ has distinct eigen values 2) $T$ has repeated eigen values 3) ...
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1answer
53 views

Proving that any unitary matrix can be diagonalised by a similar matrix

I'm having struggles with understanding important facts about spectral theorem in finite dimensional spaces. For hermitian matrices, I saw in classes that the similarity matrix that diagonalises any ...
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2answers
54 views

Show $N$ normal there is a sequence of invertible normal operators that converges to $N$.

This is a question from Conway, a course in Functional Analisys. Background is the Spectral theorem, which states that for a normal operator $N$ there is a unique spectral measure $E$ on the Borel ...
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21 views

Criteria to find a common non orthonormal basis for two linear operators

I can't find any criteria to determine, in finite dimension, if two operators has a non orthogonal common basis, for example, given two operators A and B if I check $A=A^+$ $B=B^+$ $[A, B] =0$ In ...
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92 views

Question from Axler's Linear Algebra Done Right Regarding Isometries and Normal Operators

Exercise 7.C.11 reads: "Suppose $T_1,T_2$ are normal operators on $\mathcal{L}(\mathbb{F}^3)$ and both operators have $2,5,7$ as eigenvalues. Prove that there exists an isometry $S\in\mathcal{L}(\...
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0answers
25 views

Adjoint mapping [duplicate]

I need to show the following: let $\phi$ be a linear function $V \rightarrow V$, $V$ with finite dimensions and with $\phi^*$ as adjoint function, than $\phi$ is normal if and only if there is a ...
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1answer
24 views

link between Normal operator and set of linearly independent vectors.

consider $T:\mathbb{R}^{5}\rightarrow \mathbb{R}^{5}$, $T$ is a Normal operator. prove that if $\exists w\in \mathbb{R}^{5}$ so the set :$$\left \{ w,T\left ( w \right ),T^{2}\left ( w \right ),T^{3}\...
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1answer
54 views

On the Normality of the Sum of Two Normal Operators

It is well kown that Theorem: Let $A$ and $B$ be two normal operators. If $A$ commutes with $B$, then $A + B$ is normal. Indeed the proof follows by using the Fuglede theorem since the ...
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1answer
189 views

Operator norm of a normal operator.

I know that operator norm of a self adjoint operator $T$ on $\Bbb C^n$ is given by$\|T\|=\mathrm {sup} \left \{|\left <Tx,x \right > |\ :\ \|x\|=1 \right \}$. But I just came to know that the ...
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144 views

For $2 \times 2$-matrix, $\|A^2\|=\|A\|^2$ implies that A is normal

I am reading book "A Hilbert space problem book", written by Halmos and getting some trouble. In problem 205, he claim that For two-by-two matrices an unpleasant computation proves a strong ...
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658 views

Example of normal operator on infinite-dimensional Hilbert spaces

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $F$. Definition Let $A\in\mathcal{B}(F)$. $A$ is said to be normal if $$AA^*=A^*A....
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1answer
26 views

Product between normal and hyponormal operators which commute is hyponormal

Let $A\in \mathcal{L}(E)$ be a normal operator i.e $A^{*}A=AA^{*}$. Let $B\in \mathcal{L}(E)$ be an hyponormal operator i.e. $B^*B\geq BB^*$. If $AB=BA$. Why $AB$ is hyonormal? I try to apply the ...
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139 views

Inequality involving two normal operators

Let $E$ be a complex Hilbert space, with inner product $\langle\cdot\;, \;\cdot\rangle$ and the norm $\|\cdot\|$ and let $\mathcal{L}(E)$ the algebra of all bounded linear operators from $E$ to $E$. ...
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145 views

If $\phi$ and its adjoint share a eigenvector with conjugate eigenvalue then normal

Suppose for any eigenvector $v$ of an operator $\phi:\mathbb{C}^n\rightarrow \mathbb{C}^n$ with corresponding eigenvalue $\lambda$, $v$ is also the eigenvector of $\phi^*$(the adjoint of $\phi$) with ...
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1answer
44 views

normal operator interchangeable then its adjoint also interchangeable

Let $\phi$ be a normal operator on the dimension $n$ Euclidean space, $\psi$ is a linear operator. If $\phi \psi=\psi \phi$, then $\phi^* \psi = \psi \phi^*$, where $\phi^*$ is the adjoint of $\phi$. ...
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1answer
271 views

If T is a normal operator, prove that characteristic vectors for T which are associated with distinct characteristic values are orthogonal.

The question is that If $T$ is a normal operator, prove that characteristic vectors for $T$ which are associated with distinct characteristic values are orthogonal. my proof is, let $W_i=$ ...
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1answer
290 views

A normal transformation with real eigenvalues which is not self-adjoint [closed]

I need an example of a normal transformation $T$ in an inner product space such that all its eigenvalues are real numbers, but $T\neq T^*$.
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1answer
255 views

characteristic vectors for $Τ$ which are associated with distinct characteristic values are orthogonal

I am having problem in doing the following problems: $1)$ If $T$ is a normal operator, prove that characteristic vectors for $Τ$ which are associated with distinct characteristic values are ...
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2answers
82 views

$V$ be a complex i.p.s. of dimension $n$ , $f(x) \in \mathcal P_{n-1}(\mathbb C)$ then $\exists $ linear operator $T$ on $V$ s.t. $T^*=f(T)$?

Let $V$ be a complex inner product space of dimension $n$ , let $f(x)$ be a polynomial with complex-coefficients of degree less than or equal to $n-1$ , then is it true that there exist a linear ...
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1answer
162 views

$V$ be a finite dimensional complex inner product space , $T$ be a normal operator on $V$ such that $T^*T=T^2$ , then is $T$ self-adjoint?

Let $V$ be a finite dimensional complex inner product space , $T$ be a normal operator on $V$ such that $T^*T=T^2$ , then is it true that $T^*=T$ i.e. $T$ is self-adjoint ? I only know that $T$ is ...
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269 views

Show that in a complex Hilbert space, T normal bounded linear operator, $\| T^2 \| =\| T \| ^2$

So, as a part of a problem, I've been asked to prove that if $H$ is a complex Hilbert space and $T\in L(H)$ is normal, then $\| T^2 \| =\| T \| ^2$ (Operator norm) Context: This is part (b) in a ...
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3k views

The spectral radius of normal operator

Let $H$ be a Hilbert space and $T$ be linear bounded operator in $H$. Prove that if $T$ is normal then the spectral radius of $T$, $$r(T)=\|T\|.$$ Is this TRUE?