Wiki source code of Operator Theory, fall 2007

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1 = Operator Theory, fall 2007 =
2
3 === Lecturer ===
4
5 [[Dr. Jani Virtanen>>doc:mathstatHenkilokunta.Virtanen, Jani]]
6
7 === Scope ===
8
9 10 op.
10
11 === Type ===
12
13 Advanced studies.
14
15 === Prerequisites ===
16
17 complex and real analysis, linear algebra, functional analysis
18 (or close equivalents; if in doubt discuss the requirements with the lecturer)
19
20 === Lectures ===
21
22 Periods I and II, Tuesdays and Wednesdays 2:00-4:00 in Room B322
23 The first lecture Sept. 11, 2007.
24
25 === Office Hours ===
26
27 Wednesdays 12:00-2:00
28
29 === Course Notes ===
30
31 Will be made available to download during the course.
32
33 [[Lecture notes>>url:http://www.helsinki.fi/~~jzavirta/ot2007.pdf||shape="rect"]] as of Dec. 11
34
35 === Exercises ===
36
37 Weekly problem sheets will be provided and discussed in class.
38
39 |(((
40 [[Sheet 1>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises1.pdf||shape="rect"]]
41 )))|(((
42 [[Solutions 1>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions1.pdf||shape="rect"]]
43 )))|(((
44 due Sept. 25
45 )))
46 |(((
47 [[Sheet 2>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises2.pdf||shape="rect"]]
48 )))|(((
49 [[Solutions 2>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions2.pdf||shape="rect"]]
50 )))|(((
51 due Oct. 2
52 )))
53 |(((
54 [[Sheet 3>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises3.pdf||shape="rect"]]
55 )))|(((
56 [[Solutions 3>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions3.pdf||shape="rect"]]
57 )))|(((
58 due Oct. 9
59 )))
60 |(((
61 [[Sheet 4>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises4.pdf||shape="rect"]]
62 )))|(((
63 [[Solutions 4>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions4.pdf||shape="rect"]]
64 )))|(((
65 due Oct. 16
66 )))
67 |(((
68 [[Sheet 5>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises5.pdf||shape="rect"]]
69 )))|(((
70 [[Solutions 5>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions5.pdf||shape="rect"]]
71 )))|(((
72 due Oct. 30
73 )))
74 |(((
75 [[Sheet 6>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises6.pdf||shape="rect"]]
76 )))|(((
77 [[Solutions 6>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions6.pdf||shape="rect"]]
78 )))|(((
79 due Nov. 6
80 )))
81 |(((
82 [[Sheet 7>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises7.pdf||shape="rect"]]
83 )))|(((
84 [[Solutions 7>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions7.pdf||shape="rect"]]
85 )))|(((
86 due Nov. 13
87 )))
88 |(((
89 [[Sheet 8>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises8.pdf||shape="rect"]]
90 )))|(((
91 [[Solutions 8>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions8.pdf||shape="rect"]]
92 )))|(((
93 due Nov. 20
94 )))
95 |(((
96 [[Sheet 9>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises9.pdf||shape="rect"]]
97 )))|(((
98 [[Solutions 9>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions9.pdf||shape="rect"]]
99 )))|(((
100 due Nov. 27
101 )))
102 |(((
103 [[Sheet 10>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises10.pdf||shape="rect"]]
104 )))|(((
105 [[Solutions 10>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions10.pdf||shape="rect"]]
106 )))|(((
107 due Dec. 4
108 )))
109 |(((
110 [[Sheet 11>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises11.pdf||shape="rect"]]
111 )))|(((
112 [[Solutions 11>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions11.pdf||shape="rect"]]
113 )))|(((
114 due Dec. 11
115 )))
116 |(((
117 [[Sheet 12>>url:http://www.helsinki.fi/~~jzavirta/Exercises/exercises12.pdf||shape="rect"]]
118 )))|(((
119 [[Solutions 12>>url:http://www.helsinki.fi/~~jzavirta/Exercises/solutions12.pdf||shape="rect"]]
120 )))|(((
121
122 )))
123
124 Do your homework and turn it in, and you'll receive extra points toward your final grade as follows:
125 25% completed - 1 point, 35% - 2 points, 45% - 3 points, 55% - 4 points, 65% - 5 points, and 75% - 6 points.
126
127 === Contents ===
128
129 Operator theory is no doubt a diverse area that has grown out of linear algebra and complex analysis, and is often described as the branch of functional analysis that deals with bounded linear operators and their (spectral) properties. It has developed with strong links to (mathematical) physics and mechanics, and continues to attract both pure and applied mathematicians in its vast area.
130
131 We start by quickly reviewing the core material in functional analysis, which should provide (with some extra effort) a sufficient background for those with no previous experience in functional analysis. This is followed by the study of abstract Banach algebras and spectral theory. Linear operators are considered both on Hilbert spaces and Banach spaces, and in particular we focus on compact and Fredholm operators and their index theory.
132
133 In order to study certain classes of concrete linear operators on analytic function spaces, we develop the theory of Hardy and Bergman spaces, and study spectral properties of these operators with tools introduced in the more abstract setting. This part of the course reveals the fruitful interplay between comples analysis and operator theory.
134
135 As pointed above, applications play an important role in connection with operator theory, and so we introduce as an illustration some of the most important of these, namely orthogonal polynomials and random matrices, which have received much attention lately.
136
137 In summary, we focus on the following topics:
138
139 * review of the necessary preliminaries on functional analysis
140 * Banach algebra techniques
141 * compact and Fredholm operators
142 * analytic function spaces
143 * concrete classes of linear operators
144 * the use of tools and techniques illustrated with Hankel and Toeplitz operators
145 * applications: orthogonal polynomials and random matrices
146
147 The course is suitable for those interested in analysis, mathematical physics, or applied mathematics, and provides many topics ideal for further research, e.g. for Master's thesis and beyond. This can be further discussed with the lecturer and also with other members of the analysis group.
148
149 === Literature ===
150
151 Ronald G. Douglas, Banach algebra techniques in operator theory [[MR0361893 (50 #14335)>>url:http://www.ams.org/mathscinet-getitem?mr=361893||shape="rect"]] [[MR1634900 (99c:47001)>>url:http://www.ams.org/mathscinet-getitem?mr=1634900||shape="rect"]]
152 Kehe Zhu, Operator theory in function spaces [[MR1074007 (92c:47031)>>url:http://www.ams.org/mathscinet-getitem?mr=1074007||shape="rect"]]