Title of the course: Operator Theory on Hilbert Spaces- an Introduction
Instructor: Dr. Elif Uyanık, Dr. Başak Koca, Assoc. Prof. Uğur Gül
Institution: Yozgat Bozok Ü., İstanbul Ü., Hacettepe Ü.
Dates: 19 August – 8 September 2019
Prerequisites: Yüksek lisans düzeyinde reel analiz bilgisi
I. week (Elif UYANIK)
I. Functional Analysis Preliminaries: normed spaces, Banach spaces, bounded operators
II. Hahn-Banach theorem, dual maps
III. Baire Category theorem and consequences.
IV. Inner product spaces, Cauchy-Schwarz inequality, Hilbert spaces.
V. Hilbert Spaces, Orthonormal systems, basis
II. week (Başak KOCA)
I. Banach algebras, spectrum, spectral radius, unitization, Neumann series.
II. Gelfand-Mazur theorem, Spectral mapping theorem, spectral Radius formula.
III. Closed ideals, maximal ideals, maximal ideal space.
IV. Commutative Banach algebras, Gelfand transform.
V. C*-algebras, commutative C*-algebras, Gelfand-Naimark Theorem.
III. week (Uğur GÜL)
I. Operators on Hilbert spaces, Self-adjoint, unitary, normal operators, Isometries.
II. Spectral resolution of normal operators, L\infty functional calculus.
III. Topologies on B(H): weak operator topology, strong operatör topology, Von-Neumann bicommutant theorem, Von Neumann algebras
IV. projections and projection lattices in B(H)
V. Commutative Von-Neumann algebras.
References: 1. "Introduction to Functional Analysis" by R. Meise and D. Vogt, Oxford Science Publications, 1997
2. "C*-algebras and Operator Theory" by G. Murphy, Academic Press, 1990
3. "Fundamentals of the Theory of Operator Algebras" by R. Kadison and J. R. Ringrose, Academic Press, 1983.