1
vote
1answer
48 views

Counterexample of $\text{Null}(T)^{\bot} = \text{Im}(T^{*})$

I know that $\text{Null}(T)^{\bot} = \text{Im}(T^{*})$, where $T^{*}$ means the adjoint operator of a linear operator $T$, holds when the domain of $T$ is finite-dimensional. However, the proof uses ...
3
votes
1answer
92 views

Is it true that if $A=BA^{*}A$ then $A^{*}=B^{*}AA^{*}$

I wonder is it true that for any $n \times n$ matrices $A$, and $B$.// If $A=BA^{*}A$ is true, does it imply that $A^{*}=B^{*}AA^{*}$?// I used mathematica to check the condition in 3 by 3 case, and ...
0
votes
0answers
10 views

Spectral norm of a Hadamard product

Let $F$ be the $n\times n$ DFT matrix, i.e., $F_{i,j} = \exp\left(-\tfrac{2\pi(i-1)(j-1)}{n}\imath\right)$, for $1\le i,j\le n$ and $\imath =\sqrt{-1}$. Futhermore, let $\Vert \cdot\Vert$ and $\circ$ ...
0
votes
1answer
62 views

Idempotent operators.

Apologies first. I am a physicist and my notations and rigour is probably lousy. If $P$ is an idempotent operator, $P^2 = P$, $P\neq \mathbb1$ and we have $\forall |\psi\rangle$ the relation, $P.L ...
0
votes
0answers
36 views

Canonical Forms For Matrices

In the following paper by Wedderburn what are the restrictions on the field $\mathbb F$ or on the linear application $\varphi$ that the author refers to obtain the matrix B? ...
2
votes
1answer
50 views

Relation between $A^{*}B=B^{*}A$ and $AB^{*}=BA^{*}$

Let $A$ and $B$ be two matrices. Can we say $A^{*}B=B^{*}A$ implies $AB^{*}=BA^{*}$? how about when $A$ or $B$ are normal? Any comments could be useful. Thanks.
2
votes
4answers
87 views

Square root of a Hermitian operator exists

There are a lot of questions here about square root operators, but none of them addresses the basic question of existence, and I didn't find a very beefy section in Wikipedia talking about this, so ...
2
votes
1answer
48 views

Show that $f$ is a homothety

$E$ a $\mathbb{C}$-vector space of dimension $n>2$ Let $f : E \rightarrow E$ an endomorphisme which commutes with all automorphisms of $E$. Show that $f$ is a homothety Let $\lambda$ an ...
0
votes
1answer
33 views

Prove property of adjoint: $(\mathcal{A}^{-1})^*=(\mathcal{A}^*)^{-1}$.

I'm trying to prove it like any other property of adjoint. So, I need to prove following equality: $(\mathcal{A}^{-1}x, y)=(x, \mathcal({A}^{-1})^*y)$. I know it's very basic, but how to prove this ...
1
vote
3answers
56 views

$\det A \neq 0$. Prove that $\det A^* \neq 0$.

$A$ is matrix representing operator $\mathcal{A}$. $*$ is such operator that respects following equality: $(\mathcal{A}x,y)=(x, \mathcal{A}^*y)$; (I don't know what term is used in English). ...
2
votes
1answer
29 views

Prove that $\mathcal{AB}$ is linear operator if $\mathcal{A}$ and $\mathcal{B}$ are linear operators.

It is fairly easy to determine whether $\mathcal{AB}$ is linear when we know $\mathcal{A}$ and $\mathcal{B}$ (for example, $\mathcal{Ax}=(2x_1, 3x_2-x_1)$ and $\mathcal{B}$ is something similar). But ...
0
votes
1answer
25 views

Prove that operator of mirror plane $x+z=0$ is linear and find its' matrix.

I am not familiar with term mirror plane , hence I don't know how to solve this problem. As for operator itself, maybe if I select basis $(x,0,0), (0,y,0), (0,0,z)$ then I would express $x+z$ this ...
1
vote
0answers
13 views

Is there any relation between positive definite operator and an operator that satisfies maximum principle?

Suppose $L$ is a self adjoint differential operator which satisfies maximum principle. Maximum principle: Assume that $u(x)$ satisfies $u(0)\geq 0$ and $u(1)\geq 0$. Now $L$ is said to satisfy ...
0
votes
0answers
21 views

Doubt on eigenvalues of normal operators

I'm trying to understand the solution of the following problem: $T$ is a normal operator. If $T( v)=\lambda v$, then $T^*(v)=\bar\lambda v$: The solution is: I didn't understand why we ...
0
votes
0answers
22 views

Cauchy Schwarz inequality with an operator

The standard Cauchy-Schwarz inequality is given by, $|\langle\Phi|\Psi\rangle|^2\le\langle\Phi|\Phi\rangle\langle\Psi|\Psi\rangle$ But now I'm intressted in what happens to ...
1
vote
1answer
21 views

Hermitian and Diagonal Matrix Norm inequality

I have a matrix inequality that I think is true, but I can't prove. $D_1$ and $D_2$ are diagonal matrices with non-negative entries. $M_1$ and $M_2$ are positive definite matrices. I want to show ...
0
votes
1answer
48 views

Norm of the multiplication operator

Let $f \in L^\infty[0,1].$ It is clear that the norm of the multiplication operator $M_f : g \mapsto fg$ on $L^p[0,1]$ is $\|f\|_\infty.$ What happens in the noncommutative situation? Let us ...
2
votes
1answer
30 views

Question about the operator norm on $\mathbb R^2$

So here is my question, I have to decide whether the following statement is true Let $T$ be an isomorphism on $\mathbb R^2$. Then $$\|T\|=\frac{1}{\|T^{-1}\|}$$ I am pretty sure that the statement ...
1
vote
1answer
30 views

is this equality true?

Let $\{M_\alpha\}$ be a family of closed subspaces of a Hilbert space H. Is this equality true? ${(\cap M_\alpha^\bot)^\bot}=\overline{span(\cup M_\alpha)}$. thanks for your help.
2
votes
1answer
49 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
2
votes
1answer
27 views

Fast argument to see that the dual map of a projection is a projection

If $X$ is a Banach space and $U,V$ are closed subspaces, such that $X \cong U \oplus V$, then a continuous linear map $P:X \rightarrow X$ is called a projection if $P|_U =id$ and ker(P)=V. Now we ...
2
votes
1answer
22 views

Unitary transformation between complete and orthonormal bases

I'm using the Dirac notation for vectors here, since this is a quantum mechanics question. Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the ...
0
votes
3answers
86 views

polar decomposition proof

Let $H$ be a hilbert space and $T$ a bounded linear operator on $H$. I'm trying to prove that there is a partial isometry $V$ on the closure of $Im(|T|)$ such that $T=V|T|$ and $|T|=V^*T$, where ...
1
vote
2answers
66 views

uniqueness of positive operator

Let $A,B$ be commuting positive operators on a hilbert space such that $\langle(A-B)(A+B)x,x\rangle=0$ for all $x$ in the hilbert space. Prove that $A=B$. My attempt: The above implies that $A=B$ on ...
2
votes
1answer
101 views

$\langle Tx,x \rangle=0$ proof

If $T$ is a bounded operator on a hilbert space $H$ and $\langle Tx,x \rangle=0$ for all $x$ in $H$, then $T=0$. I'm considering what we can conclude if $\langle Tx,x \rangle=0$ for all $x$ in some ...
1
vote
1answer
51 views

Operator Norm = 1

Let there be a linear map $T$ such that $T: \mathbb R^n\to\mathbb R^m$. The operator norm $\lVert \cdot\lVert_{op}$ of $T$ is then defined as the largest value of $c$ for which $\lVert T(\vec v)\lVert ...
2
votes
1answer
115 views

square root of positive operators

It $T, S$ are positive operators, do we have that $\sqrt{TS}=\sqrt{T}\sqrt{S}$? Are there any basic rules that hold for square roots of positive numbers that don't hold for positive operators?
0
votes
2answers
81 views

generalized eigenspace direct sum

Similar to the way an infinite dimensional hilbert space can be written as a direct sum of eigenspaces of a normal compact operator, I was wondering whether it can be written as a direct sum of ...
1
vote
1answer
35 views

spectral measure and integral query

I have proved the 'resolution of the identity' for a normal operator, namely that there is a unique spectral measure E such that $\int_{{\sigma}(T)} {\lambda}\,dE=T$ If (${\lambda}_{n}$) is the ...
1
vote
3answers
128 views

Borel functional calculus

For a normal operator T, we have a resolution of the identity $\int_{{\sigma}(T)} {\lambda}\,dE=T$. If $T$ is in addition compact , we have that $\sum_{n=1}^{{\infty}}{\lambda}_{n}\langle ...
1
vote
1answer
70 views

Rank of sum of projections

Let $(\varphi_j)$ be a linear independent sequence of elements of a Hilbert space, not necessarily orthogonal, but such that $$Kf := \sum_{j=1}^\infty \langle\varphi_j, f\rangle\varphi_j$$ converges ...
4
votes
1answer
81 views

Spectral decomposition of normal operator

Define $T$ from $L_{2}(R)$ into itself by $T(f)(t)=f(t+1)$. Show that $T$ is normal and finds its spectral decomposition. I've shown that $f$ is normal (in fact it's unitary) but how do I find its ...
1
vote
1answer
53 views

formula for the norm of a normal operator

In Rudin's Functional analysis, he does a theorem which shows that for a normal operator $\Vert T\Vert=\sup\left\{|\langle Tx,x\rangle|\colon \Vert x \Vert \leq 1\right\}$. Why can't $\Vert x \Vert ...
2
votes
1answer
51 views

Polar decomposition corollary

Let $T$ be a compact operator on an infinite dimensional Hilbert space. Let $|T|=(T^*T)^{0.5}$. By the polar decomposition theorem there is a partial isometry $S$ of the closure of Im$(|T|)$ such that ...
4
votes
0answers
99 views

Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
2
votes
1answer
73 views

Positive compact operator has unique square root.

Let H be a hilbert space and T be a compact positive operator so that by the spectral decomposition theorem, $T=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,e_{n}\rangle e_{n}$ where the $e_{n}$ are the ...
0
votes
2answers
20 views

Find a real matrix with eigen vectors v and v's complex conjugate so that they have different eigenvalues.

I need to find a real matrix with eigenvector v, and eigenvector v's complex conjugate, such that they will have different eigenvalue. any hints please?
0
votes
1answer
67 views

Infinite dimensional operator inverse

A is a linear operator on V and there exist a single operator B on V such that AB = I or BA = I. Prove that then A is monomorfic and epimorfic. On infinite dimensions, left and right inverses need ...
1
vote
1answer
29 views

Linear Operator identity prrof [closed]

Let A,B be invertible linear operators. Prove the identity: $B^{-1}-A^{-1}=B^{-1}(A-B)A^{-1}$
1
vote
1answer
48 views

Positive semidefinite linear operator T over a unitary space V that satisfies $T^k=I$ where $k \geq 1$ must be identity?

I got the following question in an exam I got yesterday that I didn't managed to answer: Let $V$ be a finite dimensional unitary vector space and let $T:V \to V$ be a positive semidefinite linear ...
0
votes
1answer
56 views

homomorphism or not

Let $T$ be a bounded operator on $H$ and fix a vector $x\in H$. Define $f$ on the space of polynomials in $T$ by $f(p(T))=p(x)$. Is $f$ a homomorphism? Initally I thought it obvious but the subtelty ...
1
vote
0answers
33 views

How to find all the eigenvalues of a positive operator whose eigenvectors are positive semi-defintie?

A linear operator $T:\mathcal{H}_n\rightarrow \mathcal{H}_n$ is said to be positive if $T(\mathcal{P}_n)\subset\mathcal{P}_n$ where $P_n$ is the set of positive semi-definite matrices. For a positive ...
0
votes
1answer
43 views

$T:V\rightarrow V $ is over $\mathbb{R}$ , it's matrix is $A$, $A=PDP^*$. Is it true that $A$, $D$, and $P$ are in $M_{n \times n}(\mathbb{R})$

$T:V\rightarrow V$ is over $\mathbb{R}$ and $V$ of finite dimension $n$, and I know that it is orthogonally diagonalizable. The Matrix that represents it - call it $A$ ,in orthonormal basis is ...
2
votes
2answers
176 views

$T^*T=TT^*$ and $T^2=T$. Prove $T$ is self adjoint: $T=T^*$ [duplicate]

$V$ is an inner product space of finite dimension over $\mathbb{R}$, and $T:V\to V$ a linear transformation which is normal, that is, $T^*T=TT^*$. In addition $T^2=T$. Prove $T$ is self adjoint, that ...
1
vote
1answer
66 views

If A unitary matrix and orthogonally diagonalizable why there is a basis in whichthe linear trans. matrix is diagonal?

If $A$ is a $n\times n$ unitary matrix (above the complex field) and is orthogonally diagonalizable, why does it mean that the is an orthonormal basis $\mathbb C$ in which the matrix that represent ...
1
vote
1answer
34 views

adjoint map and dual map of complex inner product space

I know (a). but I can't solve (b) and (c). Can you help me please?
1
vote
1answer
39 views

Comparison of Symmetric Operators

The Problem: There is a unitary space $(V,<.,.>)$, $D \subseteq V $ a subspace and $ A,B : V \supseteq D\to V $ are two symmetric linear operators. Show that if: $<Ax , x> $$=$ $<Bx ...
1
vote
2answers
33 views

Find the adjoint operator of $T_p$

Let $V=\mathscr{M}_n(C)$ with an inner product $\langle A,B\rangle=\mathrm{Tr}\,(AB^{*})$, $P$ be a fixed invertible matrix in $V$, and $T_P$ be the linear operator on $V$ defined by ...
0
votes
1answer
49 views

relations between two linear operators

Let $\alpha,\beta$ be linear operators on a finite dimensional vector space $V$ over field $F$. Let $\gamma=\alpha\circ\beta$ and $\delta=\beta\circ\alpha$. Prove that: (1). $m_\delta(x)$ divides ...
1
vote
1answer
73 views

Prove that if $T$ a normal linear transformation and invertible, then $T^{-1}$ is normal.

The question is: Prove that if $T$ a normal linear transformation and invertible, then $T^{-1}$ is normal. Then I have to find the spectral decomposition of $T^{-1}$. At first I tried to prove it by ...