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Math 212b Lecture 4 Stone’s theorem.

Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

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Page 1: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Math 212b Lecture 4Stone’s theorem.

Page 2: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 3: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Equibounded continuous semigroups on a Frechet space.

Page 4: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 5: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Enter the resolvent.

Page 6: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 7: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 8: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 9: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 10: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 11: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 12: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 13: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 14: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 15: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 16: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 17: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Suppose that

We know that

Page 18: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 19: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 20: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 21: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

A useful formula.

Page 22: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Example: translations on the real line.

Page 23: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 24: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 25: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 26: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Review of the Cayley transform.

Page 27: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 28: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 29: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 30: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 31: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 32: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 33: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 34: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 35: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 36: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

Page 37: Math 212b Lecture 4abel.math.harvard.edu/archive/212b_spring_05/lectures/212b0504.pdf · want to consider a one parameter family of operators Tt on F defined for all t > 0 and

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