Unlimited Borel functional calculus

The unbounded Borel functional calculus is an instrument in the mathematical theory of densely-defined self-adjoint operators . It allows such operators to be “inserted” into Borel functions , which is used, among other things, in quantum mechanics , since the self-adjoint operators are the quantum mechanical observables . This functional calculus is mathematically very complex, since dealing with densely-defined operators requires additional techniques.

initial situation

Let it be a densely defined , self-adjoint operator with a domain in a separable Hilbert space (densely defined means that it is close ). The spectrum of such an operator is the set of all , so the operator is not bijective . One can show that the spectrum of a self-adjoint operator is real. As in the case of the bounded self-adjoint operators, there is a spectral measure such that . ${\displaystyle T}$ ${\displaystyle D_{T}}$ ${\displaystyle H}$ ${\displaystyle D_{T}\subset H}$ ${\displaystyle \sigma (T)}$ ${\displaystyle \lambda \in \mathbb {C} }$ ${\displaystyle T-\lambda \,{\mathrm {id} }_{H}:\,D_{T}\rightarrow H}$ ${\displaystyle \textstyle (E_{\lambda })_{\lambda \in \mathbb {R} }}$ ${\displaystyle \textstyle T=\int _{\mathbb {R} }\lambda {\mathrm {d} }E_{\lambda }}$

If a Borel function is limited, one can ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {C} }$

${\displaystyle f(T):=\int _{\mathbb {R} }f(\lambda ){\mathrm {d} }E_{\lambda }\in L(H)}$

form, as defined due to the narrowness of a steady sesquilinear on . ${\displaystyle \textstyle (x,y)\mapsto \int _{\mathbb {R} }f(\lambda ){\mathrm {d} }\langle E_{\lambda }x,y\rangle }$ ${\displaystyle f}$ ${\displaystyle H}$

Unlimited Borel functions

Now be a Borel function that can also be unlimited. The formation of is attributed to the case of limited Borel functions as follows. Be it ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {C} }$ ${\displaystyle f(T)}$

${\displaystyle f_{n}(\lambda )={\begin{cases}f(\lambda ),&{\mbox{wenn }}|f(\lambda )|\leq n\\0,&{\mbox{sonst}}\end{cases}}}$.

Then the densely-defined operator is defined by the domain and the formula for . ${\displaystyle f(T)}$ ${\displaystyle D_{f(T)}:=\{x\in H;\,\lim _{n\to \infty }f_{n}(T)x\,{\mbox{ existiert}}\}}$ ${\displaystyle f(T)x:=\lim _{n\to \infty }f_{n}(T)x}$ ${\displaystyle x\in D_{f(T)}}$

The following can be shown for the operator defined in this way: ${\displaystyle f(T)}$

• ${\displaystyle D_{f(T)}=\{x\in H;\,\int _{\mathbb {R} }|f(\lambda )|{\mathrm {d} }\langle E_{\lambda }x,x\rangle <\infty \}}$
• ${\displaystyle \langle f(T)x,y\rangle =\int _{\mathbb {R} }f(\lambda ){\mathrm {d} }\langle E_{\lambda }x,y\rangle }$ for all ${\displaystyle x,y\in D_{f(T)}}$
• ${\displaystyle \|f(T)x\|^{2}=\int _{\mathbb {R} }|f(\lambda )|^{2}{\mathrm {d} }\langle E_{\lambda }x,x\rangle }$ for all ${\displaystyle x\in D_{f(T)}}$
• ${\displaystyle f(T)^{*}={\overline {f}}(T)}$

The functional calculus

While with the bounded Borel functional calculus for normal operators a * - homomorphism is obtained from the algebra of the bounded Borel functions , in the case of the unbounded Borel functions and densely defined self-adjoint operators one can no longer speak of a homomorphism without further ado, since the dense -defined operators do not form algebra ; the links distributive law does not apply. If namely is a densely-defined operator, the identical operator on and , then is and . In order to do justice to this fact, one must either terminate the resulting operator after each algebraic operation , this is explained in the textbook by Kadison and Ringrose with additional techniques from the theory of Von Neumann algebras given below, or as in the textbook by Dunford and Schwartz, who consider the domains of the operators. In the presentation given here, the second approach is taken. The inclusions that occur relate to the graphs of the operators, that is, one writes when is an extension of . ${\displaystyle L(H)}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle H}$ ${\displaystyle C=-B}$ ${\displaystyle A(B+C)\,=\,0}$ ${\displaystyle AB+AC=0|_{D_{A}}}$ ${\displaystyle T\supset S}$ ${\displaystyle T}$ ${\displaystyle S}$

Let it be a densely-defined self-adjoint operator on with spectral measure . Then the following rules apply to , Borel functions and Borel sets : ${\displaystyle T}$ ${\displaystyle H}$ ${\displaystyle (E_{\lambda })_{\lambda \in \mathbb {R} }}$ ${\displaystyle \alpha \in \mathbb {C} }$ ${\displaystyle f,g:\mathbb {R} \rightarrow \mathbb {C} }$ ${\displaystyle B\subset \mathbb {R} }$

• ${\displaystyle (\alpha f)(T)\,=\,\alpha f(T)}$
• ${\displaystyle (f+g)(T)\supset f(T)+g(T)}$
• ${\displaystyle D_{f(T)g(T)}\,=\,D_{(fg)(T)}\cap D_{g(T)}}$ and ${\displaystyle (fg)(T)\supset f(T)g(T)}$
• ${\displaystyle f(T)E(B)\supset E(B)f(T)}$

The following formula can be proven for the spectrum:

• ${\displaystyle \sigma (f(T))\,=\,\bigcap \{{\overline {f(B)}};\,B\subset \sigma (T){\mbox{ Borelmenge mit }}E(B)={\mathrm {id} }_{H}\}}$.

The formula is not self-evident after the construction presented here, but can be shown relatively easily. More generally, the expression for a polynomial has two possible interpretations: once as in the sense of the calculus presented above and once as in the sense of inserting it into a polynomial. One can prove that both interpretations agree, that is ${\displaystyle {\mathrm {id} }_{\mathbb {R} }(T)=T}$ ${\displaystyle p(T)}$ ${\displaystyle p(\lambda )=\alpha _{n}\lambda ^{n}+\ldots +\alpha _{1}\lambda +\alpha _{0}}$ ${\displaystyle p(T)}$ ${\displaystyle \alpha _{n}T^{n}+\ldots +\alpha _{1}T+\alpha _{0}{\mathrm {i} d}_{H}}$

• ${\displaystyle p(T)=\alpha _{n}T^{n}+\ldots +\alpha _{1}T+\alpha _{0}\mathrm {id} _{H}}$for all polynomials . ${\displaystyle p(\lambda )=\alpha _{n}\lambda ^{n}+\ldots +\alpha _{1}\lambda +\alpha _{0}}$

swell

• N. Dunford , JT Schwartz : Linear Operators, Part II, Spectral Theory. ISBN 0-471-60847-5 .
• RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras , Volume I, 1983, ISBN 0123933013