2
votes
1answer
21 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
12 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 ...
1
vote
3answers
51 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
58 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
80 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
39 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
56 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
50 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
30 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
100 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
68 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
55 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
37 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
43 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
89 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
48 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
17 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
51 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
23 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
32 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
51 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 ...
0
votes
0answers
27 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
40 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
101 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
46 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
38 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
26 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
46 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
69 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 ...
0
votes
1answer
86 views

Linear Operator and isomorphism

I wanted to be sure about the following: Let's say we have vector spaces normed spaces $X$ and $Y$ and a linear operator $T:X \rightarrow Y$. My idea was to reduce the properties that I need to show ...
0
votes
3answers
73 views

Why is this true: The only orthogonal projection that is also unitary from $\Bbb C^n$ to $\Bbb C^n$ is the identity

Can anyone explain me please how to see this statement: the only orthogonal projection that is also unitary from $\Bbb C^n$ to $\Bbb C^n$ is the Identity. how can I prove formally that? or how can I ...
1
vote
0answers
34 views

find the eigenbasis of unitary transformation

$U$ is $n\times n$ unitary matrix, with orthogonal eigenbasis $v_1, \ldots v_n$ we construct a linear transformation: $T_U(X) = XU$ with the inner product $\langle A, B \rangle = \text{tr}(A^*B)$ I ...
0
votes
5answers
123 views

Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable?

Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable? I didn't succeed to get any information about it. Could anyone explain please?
1
vote
3answers
159 views

Sum of the matrix series

Let $A\in\mathbb R^{n\times n}$ be a symmetric matrix which $0\preceq A\preceq I$ ($I$ is identity matrix), and $w_k\in\mathbb R^n$ are arbitrary certain vectors which $\|w_k\|\leq1,\,\,k=0,1,\ldots$ ...
1
vote
1answer
102 views

Prove that if transformation matrix is unitary, then the basis is orthonormal

V is a vector space with the complex field, B is an orthonormal basis of V , and C is some arbitrary basis. Prove that if the transformation matrix from basis C to B is unitary, then C is also ...
0
votes
0answers
46 views

Notation for Kronecker product of a matrix and itself?

What is the notation for the Kronecker product of a matrix and itself? In other words, is there a short-hand way I can express the following: $X⊗X$ $X⊗X⊗X$ $X⊗X⊗X⊗X$ Where $X$ is a matrix? What ...
0
votes
0answers
36 views

Are these linear operators continuous?

For every polynomial $p(t)= \sum_{k=0}^{n} a_k t^k$ we declare its norm by $||a_k||=\sum_{k=0}^{n}|a_k|$. Now, I am supposed to check whether these maps are continuous and in case that they are I ...
0
votes
4answers
43 views

Help with a 'simple' sum of linear operators and their adjoints acting on an orthonormal basis

Given an orthonormal basis $\{u_1,\cdots, u_n\}$ of a vector space $V$ I am asked to show that $$ \sum_{k=1}^n \|T^*u_k\|^2= \sum_{k=1}^n \|Tu_k\|^2 $$ for all $T\in \mathcal{L}(V)$ where $T^*$ ...
0
votes
1answer
31 views

Spectral characterization of induced operator norm

Consider $\mathbb{R}^n$ with the $l^1$ norm and the induced operator norm $\| \cdot \|$ on linear maps $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$. Can $\|T\|$ be characterized somehow by the spectrum ...
0
votes
1answer
31 views

“Almost orthogonalizing” matrices using a signature matrix

Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs). Then $||AB||_{op} \leq ...
1
vote
2answers
16 views

Hemitian operator inequality

I am trying to find two Hermitian operators $A$ and $B$ (whose representations are $2 \times 2$ complex matrices) for which neither $A \leq B$ nor $A \geq B $ holds. Note that $A \geq B$ iff ...
1
vote
0answers
58 views

Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
5
votes
1answer
98 views

Operator in the commutant with certain property

If $T$ is a bounded operator with nontrivial kernel (in my case it is actually finite dimensional kernel and the operator is quasinilpotent) acting on an infinite dimensional Banach space, can one ...
1
vote
2answers
65 views

rank of $A \otimes B$

For two matrices $A$ and $B$, what would be the rank of $A\otimes B$ as a matrix? Seems to me that $rank(A\otimes B) = rank(A)\cdot rank(B)$. But I don't see an elegant proof...
1
vote
2answers
107 views

Matrix representation of a co-domain restriction of a linear operator

Consider the finite-dimensional linear operator: $\mathcal{A}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3},$ with $Ax=y,$ $A=\left[\begin{array}{ccc} 1 & 0 & 1\\ 1 & -2 & -1\\ 0 & 1 ...
0
votes
0answers
38 views

tight frame for $\mathbb{C}^N$

I have a question to ask Prove that if $K\in\mathbb{Z}-\{0\}$, then $\{\phi_p[n]=\exp(i2\pi pn/(KN))\}_{0\leq p<KN}$ is a tight frame of $\mathbb{C}^N$, i.e. $\sum_{k}|\langle f,\phi_p\rangle ...
1
vote
1answer
70 views

What is a concrete example of a non-compact Hermitian operator on an infinite-dimensional Hilbert space whose eigenvectors do not form a complete set?

If I am not misunderstanding anything: by the spectral theorem, Hermitian operators that act upon finite-dimensional Hilbert space as well as compact Hermitian operators that act upon ...
3
votes
1answer
90 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
1
vote
2answers
102 views

How would you determine whether this sequence transformation has an inverse?

Let $T : a \mapsto b$ be a transformation of sequence $a$ to $b$ of the form $$ T(a)_m = b_m = \sum_{k=1}^{\infty} a_k e^{-i 2 \pi m / k } $$ Question. How would you go about determining if this ...