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, 1) Let L : l2 → l2 be the left орeгatol --I- --. L(a1, a2, a3, ...) = (a2; a3; a4, ...) for all a = a) Show that L is a linear and continuous operator, that is L E CL(l2). Find ||L||, the (a1, a2, a3...) E l2. operator norm of L. b) Show that each point in the open disk D = {zEC: Izl <1} is an eigenvalue of L.

, 1) Let L : l2 → l2 be the left орeгatol --I- --. L(a1, a2, a3, ...) = (a2; a3; a4, ...) for all a = a) Show that L is a linear and continuous operator, that is L E CL(l2). Find ||L||, the (a1, a2, a3...) E l2. operator norm of L. b) Show that each point in the open disk D = {zEC: Izl <1} is an eigenvalue of L.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1) Let L : l2 → 12 be the left shift operator defined by -- L(a1, a2, a3, ...) = (a2; a3, a4,...) for all a = a) Show that L is a linear and continuous operator, that is L E CL(l2). Find ||L||, the (a1, a2, a3...) E l2. %3D operator norm of L. b) Show that each point in the open disk D = {z €C: |2| <1} is an eigenvalue of L.
Transcribed Image Text:1) Let L : l2 → 12 be the left shift operator defined by -- L(a1, a2, a3, ...) = (a2; a3, a4,...) for all a = a) Show that L is a linear and continuous operator, that is L E CL(l2). Find ||L||, the (a1, a2, a3...) E l2. %3D operator norm of L. b) Show that each point in the open disk D = {z €C: |2| <1} is an eigenvalue of L.
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