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3. Let K(r, y) = xy(x+ y), 0 < x,y <1 and consider the following operator K K f(y) = | K(y, 1)f(r)dx. %3D a) Is this operator bounded as an operator acting from L'(0, 1) into itself? b) Is K bounded as an operator acting from L (0, 1) into L'(0, 1)? c) Is K bounded as an operator acting on L²(0, 1)?

3. Let K(r, y) = xy(x+ y), 0 < x,y <1 and consider the following operator K K f(y) = | K(y, 1)f(r)dx. %3D a) Is this operator bounded as an operator acting from L'(0, 1) into itself? b) Is K bounded as an operator acting from L (0, 1) into L'(0, 1)? c) Is K bounded as an operator acting on L²(0, 1)?

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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8. Problems for boundede operators 1. Let A: (2 → l² be given by 3 n+1 A(x1,., xn, ...) = In.. 2,.... n+ 2 a) Find the norm of A. b) Find the adjoint A'. c) Find the norm of A'. 2. Let A : L (0, 1) → L²(0, 1) be given by (Af)(z) = (r+t)f(t)dt. Find the adjoint A'. 3. Let K(x, y) = xy(x+ y), 0 < x, y < 1 and consider the following operator K K f(y) = | K(y, x)f(x)dx. a) Is this operator bounded as an operator acting from L'(0, 1) into itself? b) Is K bounded as an operator acting from L(0, 1) into L'(0, 1)? c) Is K bounded as an operator acting on L²(0, 1)? 4. Consider the operator A: L²(0, 1) → L²(0, 1) from problem 2. a) Is this operator bounded? If "yes", find it's norm. b) Is the open mapping theorem applicable to this operator? Why?
Transcribed Image Text:8. Problems for boundede operators 1. Let A: (2 → l² be given by 3 n+1 A(x1,., xn, ...) = In.. 2,.... n+ 2 a) Find the norm of A. b) Find the adjoint A'. c) Find the norm of A'. 2. Let A : L (0, 1) → L²(0, 1) be given by (Af)(z) = (r+t)f(t)dt. Find the adjoint A'. 3. Let K(x, y) = xy(x+ y), 0 < x, y < 1 and consider the following operator K K f(y) = | K(y, x)f(x)dx. a) Is this operator bounded as an operator acting from L'(0, 1) into itself? b) Is K bounded as an operator acting from L(0, 1) into L'(0, 1)? c) Is K bounded as an operator acting on L²(0, 1)? 4. Consider the operator A: L²(0, 1) → L²(0, 1) from problem 2. a) Is this operator bounded? If "yes", find it's norm. b) Is the open mapping theorem applicable to this operator? Why?
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