Syllabus
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Course Code: MMATH21-410 Course Name: Elective V) Advanced Functional Analysis |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Spectrum of a bounded operator: point spectrum, continuous spectrum and residual spectrum, spectral properties of bounded linear operators, the closedness and compactness of the spectrum of a bounded linear operator on a complex Banach space; further properties of resolvent and spectrum, spectral mapping theorem for polynomials. (Scope as in relevant parts of Sections 7.1 to 7.4 of Chapter 7 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) Non-emptiness of the spectrum of a bounded linear operator on a complex Banach space, spectral radius, spectral radius formula, Banach algebras, resolvent set and spectrum of a Banach algebra element, further properties of Banach algebras, spectral radius of a Banach algebra element, non-emptiness of the spectrum of a Banach algebra element. (Scope as in relevant parts of Sections 7.5 to 7.7 of Chapter 7 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) |
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| 2 | Compact linear operators on normed spaces, compactness criterion, conditions under which the limit of a sequence of compact linear operators is compact, weak convergence and compact operators, separability of range, adjoint of compact operators, Spectral properties of compact linear operators on normed spaces, eigen values of compact linear operators, closedness of the range of T , further spectral properties of compact linear operators. (Scope as in relevant parts of Sections 8.1 to 8.4 of Chapter 8 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) Operator equations involving compact linear operators, necessary and sufficient conditions for the solvability of various operator equations, further theorems of Fredholm type. Fredholm alternative. (Scope as in relevant parts of Sections 8.5 to 8.7 of Chapter 8 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) |
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| 3 | Spectral theory of bounded self-adjoint linear operators : spectral properties of bounded self adjoint operators, positive operators, projection operators and their properties. (Scope as in relevant parts of Sections 9.1 to 9.6 of Chapter 9 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) Spectral family of a bounded self adjoint linear operator, spectral representation of bounded self-adjoint linear operators, spectral theorem for bounded self-adjoint linear operators, extension of the spectral theorem to continuous functions, properties of the spectral family of a bounded self adjoint operator. (Scope as in relevant parts of Sections 9.7 to 9.11 of Chapter 9 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) |
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| 4 | Unbounded linear operators and their Hilbert adjoints, Hellinger-Toeplitz theorem, Hilbert-adjoint, symmetric and self-adjoint linear operators. Closed linear operators and closures, spectral properties of self adjoint linear operators. (Scope as in relevant parts of Sections 10.1 to 10.4 of Chapter 10 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) Spectral representation of unitary operators : Wecken’s lemma, spectral theorem for unitary operators, spectral representation for self-adjoint linear operators, multiplication and differentiation operators. (Scope as in relevant parts of Sections 10.5 to 10.7 of Chapter 10 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) |
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