1) Let L: l2 → 12 be the left shift operator defined byI --.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 = {z €C: |z| < 1} is an eigenvalue of L.c) Show that o(L) = {z €C: |z| < 1} where o(L) is the spectrum of L.d) Show that op(L) = {z € C: |z| < 1} where op(L) is the point spectrum of L.%3|

Given L:l2→l2 be the left shift operator defined by La1,a2,a3,....=a2,a3,a4,....∀a1,a2,a3,....∈l2…

| 1) L: 12 - l2 bê thể left Shift ope L(a1, a2, a3, ...) = (a2; a3, a4, ...) for all a = (a1, a2, az...) E l2. a) Show that L is a linear and continuous operator, that is L E Cl(l2). Find ||L||, the operator norm of L. b) Show that each point in the open disk D = {z €C: |z| < 1} is an eigenvalue of L. c) Show that o(L) = {z EC: |2| < 1} where o(L) is the spectrum of L. d) Show that o,(L) = {z €C: Iz| <1} where op(L) is the point spectrum of L.

| 1) L: 12 - l2 bê thể left Shift ope L(a1, a2, a3, ...) = (a2; a3, a4, ...) for all a = (a1, a2, az...) E l2. a) Show that L is a linear and continuous operator, that is L E Cl(l2). Find ||L||, the operator norm of L. b) Show that each point in the open disk D = {z €C: |z| < 1} is an eigenvalue of L. c) Show that o(L) = {z EC: |2| < 1} where o(L) is the spectrum of L. d) Show that o,(L) = {z €C: Iz| <1} where op(L) is the point spectrum of L.

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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Question

1) Let L: l2 → 12 be the left shift operator defined by
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 = {z €C: |z| < 1} is an eigenvalue of L.
c) Show that o(L) = {z €C: |z| < 1} where o(L) is the spectrum of L.
d) Show that op(L) = {z € C: |z| < 1} where op(L) is the point spectrum of L.
%3|

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