Spectral Theory of Families of Self-Adjoint Operators

'Et moi, . . . , si j'avait su comment en revenir, One service mathematics has rendered the human race. It has put common sense back je n'y serais point aile.' where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be Eric T. Bell able to do something with it. O. Hcaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics seNe as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One seIVice topology has rendered mathematical physics . . .'; 'One service logic has rendered com puter science . . .'; 'One service category theory has rendered mathematics . . .'. All arguably true. And all statements obtainable this way form part of the raison d'j~tre of this series.

Comments to the introduction. - I Families of Commuting Normal Operators. - 1. Spectral Analysis of Countable Families of Commuting Self-Adjoint Operators (CSO). - 2. Unitary Representations of Inductive Limits of Commutative Locally Compact Groups. - 3. Differential Operators With Constant Coefficients In Spaces of Functions of Infinitely Many Variables. - Inductive Limits of Finite-Dimensional Lie Algebras and Their Representations. - 4. Canonical Commutation Relations (CCR) of Systems with Countable Degrees of Freedom. - 5. Unitary Representations of The Group of Finite SU(2)-Currents on A Countable Set. - 6. Representations of The Group of Upper Triangular Matrices. - 7. A Class of Inductive Limits of Groups and Their Representations. - Collections of Unbounded Self-Adjoint operators Satisfying General Relations. - 8. Anticommuting Self-Adjoint Operators. - 9. Finite and Countable Collections of Gradedcommuting Self-Adjoint Operators (GCSO). - 10. Collections Of Unbounded CSO (Ak) And CSO (Bk) Satisfying General Commutation Relations. - Representations of Operator Algebras And Non-Commutative Random Sequences. - 11. C* -ALGEBRASU0? And Their Representations. - 12. Non-Commutative Random Sequences and Methods for Their Construction.