First of all, let me recall the main theorems about the functional calculus in order to show which properties are needed for existence and uniqueness and which properties are consequences of the definition. The continous functional calculus states the following:
Let $A\in\mathcal{L}(\mathcal{H})$ be a self-adjoint operator. Then
there is a unique map
$$\mathcal{P}_{A}:C(\sigma(A),\mathbb{C})\to\mathcal{L}(\mathcal{H})$$
$$\hspace{4.7cm}f \mapsto \mathcal{P}_{A}(f)=:f(A)$$ called
"continuous functional calculus of $A$", such that the following
properties are fulfilled for all $f,g\in C(\sigma(A),\mathbb{C})$ and
$\lambda\in\mathbb{C}$:
- $\operatorname{id}_{\sigma(A)}(A)=A$ and $1(A)=\operatorname{id}_{\mathcal{H}}$, where $1$ denotes the constant
$1$-function.
- $\mathcal{P}_{A}$ is continuous.
- $\mathcal{P}_{A}$ is a "$\star$-algebra homomorphism", which means that it is an alegbra homomorphism, i.e. $$(\lambda
f+g)(A)=\lambda f(A) + g(A)$$ multiplicative, i.e. $$(f\cdot
g)(A)=f(A)g(A)$$ and the adjoint-operation commutes with
complex-conjugation, i.e. $$\overline{f}(A)=f(A)^{\ast}.$$
To proof this, one first of all defines the map for polynomials in the obvious way and then extends the construction to all continous functions via the Weierstrass approximation theorem. Now, using the above properties as well as the definition, one can show that the continous functional calculus has some further properties:
Let $A\in\mathcal{L}(\mathcal{H})$ be a self-adjoint operator. Then
the continous functional calculus $\mathcal{P}_{A}$ has the following
additional properties:
- $\Vert f(A)\Vert = \Vert f\Vert_{\infty}$.
- If $f\geq 0$ then $f(A)$ is a positive operator.
- If $A\psi=\lambda\psi$ then $f(A)\psi=f(\lambda)\psi$.
- $\sigma(f(A))=f(\sigma(A))$.
Now, let us extend this construction to the more general case of measurable functions:
Let $A\in\mathcal{L}(\mathcal{H})$ be a self-adjoint operator. Then
there is a unique map
$$\mathcal{P}_{A}:\mathcal{B}(\sigma(A),\mathbb{C})\to\mathcal{L}(\mathcal{H})$$
$$\hspace{4.7cm}f \mapsto \mathcal{P}_{A}(f)=:f(A)$$ called
"measurable functional calculus of $A$", such that the following
properties are fulfilled for all $f,g\in C(\sigma(A),\mathbb{C})$ and
$\lambda\in\mathbb{C}$:
- $\operatorname{id}_{\sigma(A)}(A)=A$ and $1(A)=\operatorname{id}_{\mathcal{H}}$, where $1$ denotes the constant
$1$-function.
- $\mathcal{P}_{A}$ is continuous.
- $\mathcal{P}_{A}$ is a "$\star$-algebra homomorphism" as above.
- Let $(f_{n})_{n\in\mathbb{N}}\in\mathcal{B}(\sigma(A),\mathbb{C})^{\mathbb{N}}$ be a bounded sequence s.t. $f_{n}(t)\to f(t)$ for all $t\in\sigma(A)$ for some $f\in\mathcal{B}(\sigma(A),\mathbb{C})$. Then $\langle f_{n}(A)\varphi,\psi\rangle\to \langle f(A)\varphi,\psi\rangle$ for all $\varphi,\psi\in\mathcal{H}$.
In the special case of $f\in C(\sigma(A),\mathbb{C})\subset\mathcal{B}(\sigma(A),\mathbb{C})$, the uniqueness of the functional calculus is garantueed by the properties 1.-3. Now, it is important to note that for uniqueness in the general case, we also need the 4. property.
To show existence, there are several possibilities. One option is to first of all use the continuous functional calculus above to construct a set of measures. Let $f\in C(\sigma(A),\mathbb{C})$. Then we define maps $l_{\varphi,\psi}(f):=\langle f(A)\varphi,\psi\rangle$. Using this, we define maps $l_{\varphi,\psi}:C(\sigma(A),\mathbb{C})\to\mathbb{C}$, which are linear and also continuous, since
$$\vert l_{\varphi,\psi}(f)\vert\leq \Vert f\Vert_{\infty}\cdot \Vert\varphi\Vert\cdot\Vert\psi\Vert.$$
Hence, by the Riesz-Markov-Kakutani theorem , there are complex measures $\mu_{\varphi,\psi}$ such that
$$\langle f(A)\varphi,\psi\rangle=\int_{\sigma(A)}\,f\,\mathrm{d}\mu_{\varphi,\psi}.$$
Now, the main point is that the right-hand side of this equation is also meaningful for $f\in\mathcal{B}(\sigma(A),\mathbb{C})$ and hence, we can use this measures to define the functional calculus for Borel-measurable functions. More precisely, we define for all $f\in\mathcal{B}(\sigma(A),\mathbb{C})$ the map
$$(\varphi,\psi)\mapsto \int_{\sigma(A)}\,f\,\mathrm{d}\mu_{\varphi,\psi},$$
which clearly is a continuous sesquilinearform. Hence, we can use the theorem of Lax-Milgram (basically a similar statement as the theorem of Frechet-Riesz but for sesquilinear forms), which shows that there exists an operator, which we denote by $f(A)$, such that
$$\langle f(A)\varphi,\psi\rangle=\int_{\sigma(A)}\,f\,\mathrm{d}\mu_{\varphi,\psi}.$$
Using this definition, it is straight-forward to verify properties 1. - 4. above. Furthermore, it is also clear the the measurable functional calculus reduces to the continuous functional calculus in the case we take a function $f\in C(\sigma(A),\mathbb{C})\subset\mathcal{B}(\sigma(A),\mathbb{C})$.
As before, there are some immediate consequences of this definition:
Let $A\in\mathcal{L}(\mathcal{H})$ be a self-adjoint operator. Then
the continous functional calculus $\mathcal{P}_{A}$ has the following
additional properties:
- The defining property 4. can be improved: Let $(f_{n})_{n\in\mathbb{N}}\in\mathcal{B}(\sigma(A),\mathbb{C})^{\mathbb{N}}$
be a bounded sequence s.t. $f_{n}(t)\to f(t)$ for all $t\in\sigma(A)$
for some $f\in\mathcal{B}(\sigma(A),\mathbb{C})$. Then $\Vert
f_{n}(A)\psi\Vert\to\Vert f(A)\psi\Vert$ for all $\psi\in\mathcal{H}$.
- $\Vert f(A)\Vert\leq \Vert f\Vert_{\infty}$.
Hope this helps. Feel free to ask question, if something is unclear! :-)
Best Answer
Your functions satisfy $ x\delta_\lambda(x)=\lambda\delta_\lambda(x)$. So you have $$ TP_\lambda=\lambda P_\lambda. $$ This shows that any element in the range of $P_\lambda$ is a $\lambda$-eigenvector for $T$.
Conversely, if $Tv=\lambda v$, we get that $p(T)v=p(\lambda)v$ for any polynomial $p$ that has $p(0)=0$. By taking a sequence $\{p_n\}$ of polynomials such that $p_n\to\delta_\lambda$ uniformly (recall that the spectrum of $T$ is discrete), we obtain $$ \delta_\lambda(T)v=\delta_\lambda(\lambda)v, $$ which is $P_\lambda v=v$. Thus $P_\lambda$ is the projection onto the $\lambda$-eigenspace of $T$.
The eigenspaces are always closed, they are kernels of bounded operators.