0
votes
1answer
64 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
26 views

Matix column-wise multiplication operator

I'm trying to find the proper operator for a column wise multiplication. Consider $v=[v_1, v_2, ..., v_n]^T$ and $A=\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\a_{2,1} & a_{2,2} & ...
2
votes
1answer
39 views

Give a counterexample that $A$, $B$ are similar matrices in $M_{n\times n}(\mathbb{C})$ but $PAP^{-1}\neq B$ for any $P\in GL_{n}(\mathbb{R})$.

Give a counterexample that $A$, $B$ are similar matrices in $M_{n\times n}(\mathbb{C})$ but $PAP^{-1}\neq B$ for any $P\in GL_{n}(\mathbb{R})$. How to construct this example? I have obtained that of ...
6
votes
2answers
104 views

Are self-inverse operators normal?

Let $\mathcal{H}$ be an Hilbert space. Consider a bounded Operator $T:\mathcal{H}\to \mathcal{H}$. Suppose $TT=1$, does it hold, that $T^{*}T=TT^{*}$? If so, how does one show this? If not, what kind ...
4
votes
0answers
82 views

Generalisation of the trace cyclic theorem to partial traces.

The trace-cyclic theorem says that linear operators commute (or cycle) within the trace, that is $${\rm Tr}(XY) = {\rm Tr}(YX).$$ Now if $X$ and $Y$ operate on a product space $H_A \times H_B ...
0
votes
0answers
47 views

self-adjoint subalgebras of matrix algebra

Is there any classification theorem for the self-adjoint matrix subalgebras of $M_n(\mathbb{C})$ the algebra of $n \times n$ matrices over $\mathbb{C}$ ?
2
votes
1answer
60 views

A question about quotient space of $R(T^{n})$

I am reading a paper about spectral theory. The author says it is easy to see the following proposition: For $T\in L(X)$, if dim$(R(T^{d})/R(T^{d+1}))<\infty$, then $R(T^{d})$ is closed if and ...
1
vote
0answers
54 views

$\ast$-homomorphism

Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C})) $ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
7
votes
1answer
543 views

Property of partial traces

Consider the Kronecker product of $A \in M_m, B \in M_n$: $A \otimes B = \left( \begin{matrix} a_{11}B&...&a_{1m}B\\ \vdots&\ddots\\a_{m1}B&...&a_{mm}B \end{matrix} \right)$ $A ...
1
vote
1answer
92 views

Is there any operator which its spectrum corresponding to a compact set?

we know that for each operator $T$ the spectrum $\sigma(T)$ is compact. Is the converse true I mean if we have a compact set $K\neq\emptyset$, is there any operator $T$ such that $\sigma(T)=K$? I am ...
1
vote
1answer
80 views

Linear functional on Banach algebra

Let $A$ be the space of all matrices of the form $\begin{pmatrix} a & b \\0 & a\end{pmatrix}$, $2\times2$ over complex field. Then the spectrum of any element of $A$ comes out to be $\{a\}$. I ...
1
vote
1answer
71 views

Operator inequalities: $0 \leq A \leq B \Rightarrow Tr(A^p) \leq Tr(B^p)$?

It is trivial to show that $0 \leq A \leq B \Rightarrow Tr(A^2) \leq Tr(B^2)$, but does this generally hold for all $p >$ 2 as well?
1
vote
1answer
45 views

Kernel inclusion implies factorization

I have a question whether a certain fact is true for arbitrary operators on a Hilbert space. Namely, consider Hilbert spaces $H,K$, an operator $A\in B(H)$ and another $B\in B(H,K)$. Moreover, assume ...
0
votes
1answer
83 views

Is the cone of squares in a Jordan algebra a cone?

Let $A$ be a finite-dimensional Jordan algebra over $\mathbb{R}$, i.e. a finite-dimensional real vector space with a commutative bilinear product $\circ: A \times A \rightarrow A$ satisfying $(a^2 ...
1
vote
0answers
101 views

Time-derivative of an operator

Would I be right in thinking that the operator $$\hat O'(t)$$ is different from the operator $$D\hat O(t)$$ where $D={d\over dt}$, since when acting on a function $f$, the second corresponds to $$\hat ...
2
votes
0answers
84 views

Sign problem with Poisson brackets

I am wondering if anyone could explain to me either why my method is not valid or point out where I have made an algebraic slip. I have been looking at this for a long time, to no avail. Let $\{\cdot ...
5
votes
2answers
154 views

Duals via a Bilinear map

Let $E$ and $F$ be normed vector spaces. Then if $B$ is a bounded bilinear form on $E \times F$ then every $y \in F$ defines a bounded linear functional $f_y$ where $f_y(x)=B(x, y) \forall x \in E$. ...
4
votes
3answers
162 views

Lower bound for $\|A-B\|$ when $\operatorname{rank}(A)\neq \operatorname{rank}(B)$, both $A$ and $B$ are idempotent

Let's first focus on $k$-by-$k$ matrices. We know that rank is a continuous function for idempotent matrices, so when we have, say, $\operatorname{rank}(A)>\operatorname{rank}(B)+1$, the two ...
1
vote
1answer
153 views

An upper bound for $\|(\lambda-A)^{-1}\|$?

Let $A$ be a k-by-k matrix and $\sigma(A)$ its spectrum, or the collection of eigenvalues of $A$. If we know $\lambda\notin\sigma(A)$, then $\lambda$ is at a positive distance to all points in the ...
1
vote
1answer
282 views

Orthogonal Projectors and Operators

If we have a projection T on a finite-dimensional inner product V, and we know that ||Tx|| = ||x||, can we conclude that T is an orthogonal projection? The equality with the norms is enough to ...
1
vote
0answers
42 views

Does the locality or non-locality of operators imply matrix structure?

I understand that an operator, $\hat{O}$, is said to be non-local if $$b(x)=\hat{O}a(x)=\int dx'O(x,x')a(x')$$ that is, to find $b(x)$ at aparticular value of $x$, we need to know ...
2
votes
3answers
366 views

Eigenvalues of operators

I have a linear operator $T$ which acts on the vector space of square $N\times N$ matrices in this way: $T(A)=0.5(A-A^\mathrm{t})$ ($A^\mathrm{t}$: the transpose of a matrix $A$). I need to prove ...