Articles of operator theory

proving that $\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$

In proof of Theorem 6.4.1 of Auslender’s book about asymptotic cones, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that $\epsilon^{-1}(C-\text{rge}\,A)\subset\text{aff}\,(C-C)$, that I can’t verify it. It’s clear that $\epsilon^{-1}(C-\text{rge}\,A)\subset\epsilon^{-1}(\text{aff}\,(C)-\text{aff}\,(C))$, so if we let $\omega\in\epsilon^{-1}(\text{aff}\,C-\text{aff}\,C)$, then $$\omega=\sum_{i=1}^m\epsilon^{-1}\lambda_iu_i-\sum_{i=1}^m\epsilon^{-1}\mu_iv_i=\sum_{i=1}^m\lambda_i(\epsilon^{-1}u_i-\epsilon^{-1}\frac{\mu_i}{\lambda_i}v_i),u_i,v_i\in C,\sum_{i=1}^m\lambda_i=\sum_{i=1}^m\mu_i=1$$ which is in $\text{aff}(C-C)$ iff $\epsilon^{-1}u_i,\epsilon^{-1}\frac{\mu_i}{\lambda_i}v_i\in C$, but we just have that $C$ is a convex […]

Consequence of the polarization identity?

Here is a proof which I do not fully understand. Theorem : Let $H$ be a Hilbert space. A continuous linear map $T : H \rightarrow H$ is self-adjoint (hermitian) if and only if $$\big\langle T(x), x\big\rangle \in \mathbb{R},~~~~~(\forall x \in H)$$ Proof : ($\Rightarrow$) $$ \overline{\big\langle T(x), x\big\rangle} = \big\langle x, T(x)\big\rangle = \big\langle […]

The strong topology on $U(\mathcal H)$ is metrisable

The strong operator topology on a Banach space $X$ is usually defined via semi-norms: For any $x \in X$, $|\cdot|_x: B(X) \to \mathbb R, A \mapsto \|A(x)\|$ is a semi-norm, the strong topology is the weakest/coarsest topology which makes these maps continuous. Alternatively it is generated by the sub-base $\left\{B_\epsilon(A;x)=\{B\in X \mid |B-A|_x<\epsilon\}\phantom{\sum}\right\}$. If we define […]

Spectrum of Laplacian on Half line. $\left [0, \infty \right)$

I would like to calculate the spectrum of Dirichlet and Neumann Laplacian of the domain $\left [0,\infty \right)$. To be precise, Define the Operator $T$ on $L^2\left[0,\infty\right)$ as $Tf=-f”$ and $D(T)=H^2\cap H_0^1$. And $S$ on $L^2\left[0,\infty\right)$ as $Sf=-f”$ and $D(T)=H^2 \cap \left \{f\in H^2 | f'(0)=0 \right \}$. Find out the Spectrum of T and […]

Do there exist bounded operators with unbounded inverses?

I have just been introduced to the concept of invertibility for bounded linear operators. Specifically, we defined a bounded operator $A$ to be invertible if there exists a bounded $A^{-1}$ which is its right and left inverse, i.e. $AA^{-1}=\mathrm{id}_{\mathrm{Im}A},A^{-1}A=\mathrm{id}_{\mathrm{Dom}A}$. So I was wondering: is the requirement of boundedness (or equivalently of continuity) of the inverse […]

Positive bounded operators

Let $A,B$ be positive self-adjoint bounded operators and $\lambda >0$ then I want to show that if $$A-B \ge 0 $$ that is $\langle x,(A-B)x \rangle \ge 0$ we have that the resolvents (whose existence is clear) satisfy $$(A+\lambda I)^{-1}-(B+\lambda I)^{-1} \le 0,$$ i.e. exactly the opposite relation. Although this is intuitively clear, I got […]

Norm of the sum of projection operators

Is it true that $$|| a R+b P||\leq\max \{|a|,|b|\},$$where $a$ and $b$ are complex numbers and $P,R$ are (orthogonal) projection operators on finite-dimensional closed subspaces of an infinite-dimensional hilbert space ? In finite dimensions this would be true, since a projection there has always norm one.

Spectral Measures: Spectral Spaces (II)

Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote its probability measures by: $$\nu_\varphi(A):=\|E(A)\varphi\|^2$$ Introduce the pure-point space: $$\mathcal{H}_0(E):=\{\varphi:\exists\#\Lambda_0\leq\aleph_0:\nu_\varphi(\Lambda_0)=\nu_\varphi(\Omega)\}$$ Construct its normal operator: $$\varphi\in\mathcal{D}(N):\quad\langle N\varphi,\chi\rangle=\int_\mathbb{C}\lambda\mathrm{d}\langle E(\lambda)\varphi,\chi\rangle\quad(\chi\in\mathcal{H})$$ Regard its eigenspace: $$\mathcal{E}_\lambda=\{\varphi:N\varphi=\lambda\varphi\}:\quad\mathcal{E}(N):=\cup_{\lambda}\mathcal{E}_\lambda$$ Then one has: $$\mathcal{H}_0(E)=\overline{\langle\mathcal{E}(N)\rangle}$$ How to prove this? Reference This thread is related to: Spectral Spaces (I)

Neumann series expansion for the resolvent

If $T$ is an operator on $l_2$, and $\lambda>r(T)$ (where $r(T)$ denoted the spectral radius of $T$), the resolvent $(\lambda I-T)^{-1}$ can be expanded as $$ (\lambda I-T)^{-1}=\frac{1}{\lambda}I+\frac{1}{\lambda^2}T+\frac{1}{\lambda^3}T^2+\dots $$ If we fix some $x\in l_2$, and denote $y:=(\lambda I-T)^{-1}x$ the above expansion implies that $y$ is in the closed span of $(T^nx)_{n=0}^{\infty}$. Is $y$ in […]

On Fredholm operator on Hilbert spaces

Let $u: H \to H’$ be a continuous linear operator and $H,H’$ be Hilbert spaces. Let $u^\ast$ denotes its adjoint. By definition, an operator $u$ is called Fredholm if and only if $\ker u$ has finite dimension and $\mathrm{im}(u)$ has finite codimension. My book states that “…$u$ is Fredholm if and only if $u(H)$ is […]