The tag has no usage guidance.

491 questions
Filter by
Sorted by
Tagged with
1 vote
70 views

• 169
16 views

### Infinite dimensional version of Cauchy interlacing theorem

One formulation/application of the Cauchy interlacing theorem in finite dimensions could be written as follows. Let $X$ be a finite dimensional, real Hilbert space. Let $Q \colon X \to X$ be a ...
• 3,774
38 views

31 views

• 545
57 views

### Show that operator is self-adjoint

Show that the operator B in a space of $2 \times 2$ real matrices is self-adjoint $$BX = X \begin{pmatrix} 1 & 2 \\ 2 & 4 \\ \end{pmatrix}$$ I attempted to apply the theorem that an ...
• 143
28 views

### Orthogonal Projection in an Enlarged Hilbert Space

Let $(H, \langle \cdot, \cdot \rangle_H)$ and $(U,\langle \cdot, \cdot \rangle_U )$ be Hilbert Spaces such that $H$ embeds into $U$. Let $M$ be a closed subspace of $U$, and define $\mathcal{P}$ to be ...
1 vote
79 views

24 views

### Inequality for inverse of an unbounded self-adjoint operator

Given an unbounded self-adjoint operator $A$ on some Hilbert space $\mathcal{H}$, and $\mu$ a non zero real number: I want to show that \lVert (\mathbf{A} + i\mu \mathbf{I})^{-1} \...
27 views

### Product of a Hilbert-Schmidt operator and a Kato perturbation of a self-adjoint operator

Let $\mathcal{H}$ be a Hilbert space, $A:\mathcal{H}\rightarrow\mathcal{H}$ a Hilbert-Schmidt operator on it, and $H_0:\mathcal{D}(H_0)\rightarrow\mathcal{H}$ an unbounded self-adjoint operator on it. ...
• 391
84 views

• 428
120 views

• 1,285
1 vote
32 views

### When is the operator associated to a sesquilinear form normal?

I am self-studying from Linear Algebra by Hoffman and Kunze. In chapter 9, it is stated that every sesqui-linear form $f$ on a finite-dimensional inner product space $(V,(\cdot|\cdot))$ can be ...
36 views

### $0 ≤ A ≤ I.$ $\iff$ ⟨ψ|A|ψ⟩ ∈ [0, 1] for every unit vector $|ψ⟩ ∈ H$

Given two operators $A$and$B$, where $A ≤ B$ means the operator $B − A$ is positive semidefnite. (i) $0 ≤ A ≤ I.$ (ii) ⟨ψ|A|ψ⟩ ∈ [0, 1] for every unit vector $|ψ⟩ ∈ H$ are equivalent I am having ...
• 589
1 vote
40 views

### Sum of the spectra of two s-a operators

I am asking something a little more general than my previous question. It seems "trivial" but I cannot find this statement anywhere and I have self-doubts. Let $L_1$ and $L_2$ be two self-...
• 1,139
1 vote
36 views

### Find the norm of the following operator

Let $H = L^2([-1,1],\lambda)$ with $\lambda$ the Lebesgue measure. Consider $$T: H \to H: f \mapsto Tf$$ such that $(Tf)(x) = x^2f(x)$. Compute the norm of $T$. Seems easy enough, but I am only able ...
175 views

### Question about inclusion of domains of unbounded operators on a Hilbert space

Assume that T be a self-adjoint operator on the Hilbert space $L^2(0,1)=\{f:\int_0^1 |f(x)|^2 dx<\infty\}$ with the domain $D(T)$ satisfying $C^\infty_c(0,1)\subset D(T)$ (here $C^\infty_c(0,1)$ ...
• 797
115 views

1 vote
94 views

### Is the composition of positive, self-adjoint operators on a Hilbert space positive?

Let $A,B$ be self-adjoint, bounded, linear operators on a Hilbert space $H$ over $\mathbb{R}$ such that $\langle Ax,x\rangle\geq 0$ and $\langle Bx,x\rangle\geq 0$ for all $x\in H$. Does it also hold ...
• 11
1 vote
19 views

### If $P$ is a projection and $D$ is of trace class, is it true that $\mathrm{tr}(PDP)=\mathrm{tr}(DP)$?

According to Varadarajan$^{1}$, in a separable Hilbert space, If $A$ is of trace class and $B$ is any bounded operator, $AB$ and $BA$ are of trace class; and moreover, $tr(AB) = tr(BA)$. Orthogonal ...
• 2,065
54 views

I'm currently learning linear algebra, and I was confused by these two terminologies. It seems that adjoint is the tranpose of cofactor matrix, and a self-adjoint operator has a matrix representation ...
31 views

• 1,139
99 views

### Laplace-Beltrami operator self-adjoint in context of Markov processes and their infinitesimal generators

Let $d\vec{X} = \vec{\mu}(\vec{X})dt +\sigma(\vec{X})d\vec{B}_t$ be a multidimensional SDE. It has infinitesimal generator \mathscr{L}f = \vec{\mu}^T \nabla f + \frac12 \operatorname{Tr}(\Sigma \...
• 1,092