All Questions
Tagged with operator-theory linear-algebra
956
questions
0
votes
1
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30
views
Boundedness of a matrix operator in a norm
I would like to ask a simple question.
How do I show that a matrix is a bounded linear operator, for example this matrix
$$A=\begin{bmatrix}
t~~0\\
0~~\frac{1}{\sqrt{t}}
\end{bmatrix}$$
I know that ...
- 537
0
votes
0
answers
8
views
Almost orthogonal operators after a relative scaling
If two positive operators $Q_1$ and $Q_2$ with unit $\ell_1$ norm are almost orthogonal: $\parallel Q_1 - Q_2 \parallel_1 \geq 2 -\epsilon$, then what can we say about the operators $Q_1$ and $c Q_2$, ...
- 1
-1
votes
1
answer
27
views
Stability of Subspaces under a Linear Map in Direct Sum Decomposition
Consider the vector spaces $D_1$, $D_2$, $D$ and $X$ such that $D\subset X$ and $D=D_1\oplus D_2$.
Furthermore, suppose that $L:X\longrightarrow D$ is a linear map such that $D_1$ is stable under $L$...
- 87
3
votes
2
answers
37
views
Kernel dimension is preserved under uniform convergence of bounded operators on a Hilbert space
I hope you can help me with the following question. Consider a sequence $C_{n}$ of bounded operators on an infinite-dimensional separable complex Hilbert space $\mathcal{H}$ that converges uniformly ...
- 43
0
votes
0
answers
36
views
Operators that preserve RKHSness?
Are there any results on operators that preserve the reproducing property?
As an example, orthogonal projection preserves this property (and maps the reproducing kernel to the reproducing kernel as a ...
0
votes
0
answers
55
views
Forming real symmetric positive semidefinite matrices from complex matrices.
Let $Q \in \mathbb{C}^{n\times n}$ be any matrix. When can we say that the matrix $A=Q^{t}Q$ (where $Q^{t}$ denotes the transpose of a matrix) is a real symmetric positive semidefinite matrix?
Write $...
- 11
8
votes
1
answer
265
views
Bounding $\|A^r - B^r\|$ for $r \geq 1$.
Let $A$, $B$ be positive semi-definite self-adjoint matrices. Is it true that, for $r \geq 1$, $r \in \mathbb{R}$,
$$
\|A^r - B^r\| \leq rc^{r-1}\|A - B\|
$$
where $\|\cdot\|$ denotes the operator ...
- 685
0
votes
1
answer
32
views
Is $R(w\otimes x+z\otimes y)=sp\{w,z\}$ when $x, y$ are linearly independent?
Let $H$ be a Hilbert space and let $f\otimes g$ ($f,g\neq 0$) denotes the rank-one operator defined by $(f\otimes g)h=\langle h,g\rangle f$. I know that $R(f\otimes g)=sp\{f\}$.(R denotes the range.)
...
- 573
0
votes
1
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74
views
If $A^*A=A^*B=B^*A=B^*B$, prove that $A=B$.
Suppose that $H$ is a Hilbert space and $A,B\in B(H)$ are such that $A^*A=A^*B=B^*A=B^*B$,then $A=B$.I'm having difficulty trying to read a proof that solves this problem.
First, the author decomposes ...
- 573
1
vote
1
answer
30
views
states and density matrices, compactness
I just started reading about the $C^*$-algebras so I'm hoping that what I'm writing won't be awfully wrong. I had a doubt regarding the connection between states in an algebra and density operators in ...
- 75
1
vote
1
answer
35
views
Representing Vector Norm Through Operator Norms
Let $V$ be a vector space over $\mathbb R$ and $v\in V$. We endow $V$ with a norm $\|\cdot\|_V$ and $\mathbb R$ with the standard norm. I am trying to prove
\begin{equation}
\sup \{\lvert Lv\rvert \...
- 490
2
votes
1
answer
86
views
isometries and unitary operators, Specht theorem
I was looking at the properties of the trace operator $\operatorname{tr}$, in particular the properties of the trace regarding isometric conjugation.
We say that $T\in \mathcal{H}$ is an isometry if $...
- 75
1
vote
1
answer
23
views
Vector Norm $\vert\vert v\vert\vert_V$ expressed as supremum of $\vert Lv\vert$ over all bounded operators L with $\vert\vert L\vert\vert_{op}\leq 1$
Let $V$ be a normed vector space with norm $\vert\vert\cdot\vert\vert_V$. How can I show that for all $v\in V$ we have
$$\vert\vert v\vert\vert_V = \sup\{\vert Lv\vert \:\colon\: L\in \text{Hom}(V,\...
- 567
1
vote
0
answers
23
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$\|B\|_{op} \le \|A\|_{op}$ when $B_{i,j} = v_i^TAv_j$
I'm currently stuck to the following statement.
Let $A\in\mathbb R^{n\times n}$ be a diagonal matrix whose diagonal elements are only 0 or 1, and let $B$ be $n\times n$ matrix such that $B_{i,j} = ...
- 769
1
vote
1
answer
120
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Proof by induction in the Baker–Campbell–Hausdorff formula
Whilst studying Commutators, I stumbled across this post in which the Baker–Campbell–Hausdorff formula is being proven. In one of the answers, it is stated that a certain step can be proven by ...
- 389