0
votes
0answers
37 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
0answers
35 views

Simple Modules over the Weyl Algebra

Let $k$ be a field of characteristic zero and let $A_1=k\langle x,y| \, xy-yx=1 \rangle$ be the Weyl algebra. Is there a (more or less explicit) possibility of writing down all simple modules over ...
1
vote
1answer
57 views

Question about the essential spectrum of a negative difinite operator

please on an infinite dimensional Hilbert space how to difine the essential spectrum of an operator which is negative definite ??? Please help me Thank you.
0
votes
0answers
42 views

H P S class operators and their inequalities

First few definitions: $A \in I(K)$ iff $A$ is isomorphic to some member of $K$ $A \in S(K)$ iff $A$ is a subalgebra of some member of $K$ $A \in H(K)$ iff $A$ is a homomorphic image of some ...
2
votes
1answer
54 views

Inequality of Class operators H S and P

First few definitions: $A \in I(K)$ iff $A$ is isomorphic to some member of $K$ $A \in S(K)$ iff $A$ is a subalgebra of some member of $K$ $A \in H(K)$ iff $A$ is a homomorphic image of some ...
1
vote
0answers
19 views

Differential Realizations of certain algebras

I'm a first year graduate student in Mathematical Physics, and I am trying to generalise a certain method involving the so-called "Differential realizations" of certain algebras. The problem I'm ...
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
0answers
46 views

A question about tensor product of algebras of compact operators. [duplicate]

Let $\cal{H}$ be a separable Hilbert space and $\cal{K(\cal{H})}$ the algebra of compact operators acting on $\cal{H}$. Then $$\cal{K(\cal{H})}\otimes\cal{K}(\cal{H})\cong\cal{K}(\cal{H}\otimes H).$$ ...
2
votes
0answers
111 views

Does ternary operations have associative property?

Binary Operation is a function. Right? We know that all Binary operations have associative property. They must be either associative or non-associative. The condition is : $$(a*b)*c = a*(b*c)$$ ...
0
votes
1answer
353 views

Matrix norm of a normal matrix

A normal matrix defined over a complex vector space has the property, that $\|A\|_2$ is its largest eigenvalue and now I was wondering whether this is also true for matrices defined over the real ...
1
vote
1answer
46 views

$(P\Lambda P^{-1}=T^2)~\implies~(\exists \Lambda'~\text{s.t.}~T=R\Lambda' R^{-1})$: $\;P,R\;$ Unitary Matrices

Let $T$ be a linear operator such that the operator $T^2$ is diagonalizable. Is $T$ necessarily diagonalizable?
0
votes
3answers
96 views

Diagonalizable Operators: An Operational Extension

Let $T$ be a diagonalizable operator on a vector space $V$. Prove that the operator $$a_nT^n + a_{n-1}T^{n-1}+\cdots+a_1T+a_0 Id_V$$ on $V$ is also diagonalizable for any scalars $a_1, ...
8
votes
4answers
1k views

Are all algebraic commutative operations always associative? [duplicate]

I know that there are many algebraic associative operations which are commutative and which are not commutative. for example multiplications of matrices as associative operation is not commutative. I ...
1
vote
1answer
463 views

Commutator relationship proof $[A,B^2] = 2B[A,B]$

I'm trying to find the condition necessary for this commutator relationship equality: $$[A,B^2]=2B[A,B]$$ So far I've done this: \begin{align*} [A,B^2] & = B[A,B] + [A,B]B \\ ...
2
votes
0answers
224 views

Eigenfunctions/Invariance of generic convolution operators

Suppose we are given a convolution operator $$ \mathcal{K}[f\,](t):=\int K(t-s)f(s)ds $$ acting on $f\in H_1$ where $H_1$ is a vector space with orthonormal basis $\{\phi_n(t)\}_{n=0}^{N-1}$. If ...
1
vote
1answer
112 views

How many $1$'s could there be in this sequence? Matrix, operator?

For each $(i,j)\in \mathbb{N}^2$, $a(i,j)=1$ or $0$, and 1) $a(i,i)=0$ for all $i$; 2)for fixed $i$, there is at most one $j$ such that $a(i,j)=1$. Suppose we know that there is a finite $\kappa$ such ...
2
votes
1answer
258 views

Tensor product of Hilbert Spaces

I am following this link under "definitions" I need to see why the suggested inner product on the pre-Hilbert space $H_1$ tensor $H_2$ is well defined. Recall that the fundamental tensors are a ...
1
vote
1answer
113 views

Properties of an algebra of operators that are invariant under scalar multiplication

I am thinking about the invariant subspace problem or some related problems like the almost invariant half-space problem. In this type of problems one has the following If a statement holds for an ...