I'm reading a survey paper on model spaces, there is written "the study can therefore be restricted to CNU operators, which with some additional hypotheses obtain concrete a functional model" What is a "functional model"?
Googleing donst help since this seem to be a well used term in alot of subjects and I cant understand it from Nagy and Foias book which it in the references. I have and idea that it would be some kind of functional calculus but Im not really sure.
I understand that one can "model" some contractions as restrictions of the shift to $\phi H^{2}$ where $\phi$ is inner but I fail to see how this would be related to some functional calculus.
Suppose $\|A\| < 1$. You can take limits of $rA$ as $r\uparrow 1$ for a general contraction $A$, which makes the following construction generally useful for any $A\in\mathcal{L}(\mathcal{H})$ for $\|A\| \le 1$, though the limiting case involves an positive operator measure on the unit circle $\mathbb{T}=\partial D$, which is the boundary of the unit disk $D$.
If $f$ is holomorphic on a neighborhood of the closed unit disk $D$, then, for $|z| < 1$, the following reflection argument comes from the Poisson integral representation: \begin{align} f(A) & = \frac{1}{2\pi i}\oint_{\partial D}\frac{f(w)}{w I-A}dw \\ & = \frac{1}{2\pi i}\oint_{\partial D}f(w)\left[\frac{1}{wI-A}-\frac{A^*}{w A^*-I}\right]dw \\ & =\frac{1}{2\pi i}\oint_{\partial D}f(w)\left[\frac{1}{wI-A}+\frac{\overline{w}A^*}{\overline{w}I-A^*}\right]dw \\ & =\frac{1}{2\pi i}\oint_{\partial D}f(w)\frac{1}{\overline{w}I-A^*}\left[(\overline{w}I-A^*)+\overline{w}A^*(wI-A)\right]\frac{1}{wI-A}dw \\ & =\frac{1}{2\pi i}\oint_{\partial D}f(w)\frac{1}{\overline{w}I-A^*}(I-A^*A)\frac{1}{wI-A}\overline{w}dw. \end{align} Let $\sigma$ be the normalized Lebesgue measure on the unit circle, and let $|B|=\sqrt{B^*B}$. Then $$ f(A) = \int_{\partial D}f(w)\left|\sqrt{I-A^*A}\frac{1}{wI-A}\right|^2d\sigma(w). $$ In particular, for $n=0,1,2,3,\cdots$, \begin{align} A^n & = \int_{\partial D}w^n\left|\sqrt{I-A^*A}\frac{1}{wI-A}\right|^2d\sigma(w) \\ (A^*)^n & = \int_{\partial D}\overline{w}^n\left|\sqrt{I-A^*A}\frac{1}{wI-A}\right|^2d\sigma(w). \end{align} You can define a new Hilbert space $\mathcal{K}$ as the closure of polynomials in $w,\overline{w}$ on $T$ with vector coefficients. For such a polynomial, $$ p(w,\overline{w})=p_{-n}\overline{w}^n+\cdots+p_{-1}\overline{w}+p_0+p_1 w+\cdots+p_{n}w^{n},\;\; p_j \in \mathcal{H}, $$ define an inner product through the norm $$ \|p\|_{\mathcal{K}}^2 = \int_{\partial D}\left\|\,\left|\sqrt{I-A*A}\frac{1}{wI-A}\right|p(w,\overline{w})\,\right\|^2d\sigma(w). $$ Define a map $V : \mathcal{H}\rightarrow\mathcal{K}$ so that $Vx$ is the constant polynomial $p(w,\overline{w})=x$. Then $$ \|Vx\|_{\mathcal{K}}=\|x\|_{\mathcal{H}},\;\;\; x\in\mathcal{H}. $$ Let $M : \mathcal{K}\rightarrow\mathcal{K}$ be defined by $(Mp)(w,\overline{w})=wp(w,\overline{w})$. Then $M$ is unitary, and $$ (M^n Vx,Vy)_{\mathcal{K}}= (A^nx,y)_{\mathcal{H}},\;\;\; x,y\in \mathcal{H} \\ V^*M^nV = A^n,\;\;\; n=0,1,2,3,\cdots. $$