Event Title

The Volterra Operator Is Not Supercyclic

Presenter Information

John Dixon, Colby CollegeFollow

Location

Diamond 323

Start Date

1-5-2014 3:00 PM

End Date

1-5-2014 4:00 PM

Project Type

Presentation

Description

First studied by Vito Volterra during the 1920s and 30s, the Volterra Operator is easy to understand but still is interesting to study. The Volterra Operator is a gadget that takes in a nice function and produces another nice function using the Fundamental Theorem of Calculus. While its definition is simple the Volterra Operator possesses many interesting properties and serves as an elementary example of more complex gadgets. Volterra Operator is Cyclic Operator, however it can be shown that it fails to be Supercyclic. For an operator to be Cyclic there must exist a vector whose orbit, image under repeated applications of the operator, has a dense linear span in the range. Being Supercyclic is a stronger property that requires the existence of a vector whose orbit up to scaling is dense in range. The talk will outline a proof that the Volterra Operator is not Supercyclic and along the way give several other important results about the Volterra Operator.

Faculty Sponsor

Leo Livshits

Sponsoring Department

Colby College. Mathematics and Statistics Dept.

CLAS Field of Study

Natural Sciences

Event Website

http://www.colby.edu/clas

ID

514

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May 1st, 3:00 PM May 1st, 4:00 PM

The Volterra Operator Is Not Supercyclic

Diamond 323

First studied by Vito Volterra during the 1920s and 30s, the Volterra Operator is easy to understand but still is interesting to study. The Volterra Operator is a gadget that takes in a nice function and produces another nice function using the Fundamental Theorem of Calculus. While its definition is simple the Volterra Operator possesses many interesting properties and serves as an elementary example of more complex gadgets. Volterra Operator is Cyclic Operator, however it can be shown that it fails to be Supercyclic. For an operator to be Cyclic there must exist a vector whose orbit, image under repeated applications of the operator, has a dense linear span in the range. Being Supercyclic is a stronger property that requires the existence of a vector whose orbit up to scaling is dense in range. The talk will outline a proof that the Volterra Operator is not Supercyclic and along the way give several other important results about the Volterra Operator.

http://digitalcommons.colby.edu/clas/2014/program/209