Q6/ Let I₁, I₂ be two bounded operators on H with T²=T₁=T₁ T2 = I₁=I₂₁. Can we say that if (T₁₂-II₂+1₂)² = (₁-T₂ +T₂) and ker(T₁ – T₂T₂ +T₂₁) = R(T₁ −₂+₂), then ₁₂ = LI? (Proof or counter-example with details).
Q6/ Let I₁, I₂ be two bounded operators on H with T²=T₁=T₁ T2 = I₁=I₂₁. Can we say that if (T₁₂-II₂+1₂)² = (₁-T₂ +T₂) and ker(T₁ – T₂T₂ +T₂₁) = R(T₁ −₂+₂), then ₁₂ = LI? (Proof or counter-example with details).
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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Transcribed Image Text:Q6/
Let I₁, I₂ be two bounded operators on H with T²=₁=T₁ T₂ = I₁=I₂. Can we say that if
(I-II₁₂+1₂)² = (I-IT₂+T₂) and ker(T₁ − T₁T₂ +T₂) = R(T₁ −₂+₂), then ₁₂ = ₂₂?
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(Proof or counter-example with details).
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