3. QUESTION:
Show that for bounded linear operator A,B on a
normed space, 𝐴𝐵 ≤ 𝐴 𝐵 .Is it true that 𝐴𝐵 = 𝐴 𝐵 ?
Solution:
For every y,
𝐴𝑦 ≤ 𝐴 𝑦 .
Hence
𝐴(𝐵𝑥) ≤ 𝐴 𝐵𝑥 ≤ 𝐴 𝐵 𝑥 .
𝐴𝐵 = 𝑠𝑢𝑝 𝑥 =1 𝐴𝐵(𝑥) ≤ 𝑠𝑢𝑝 𝑥 =1 𝐴 𝐵 𝑥
≤ 𝐴 𝐵
This implies that
𝐴𝐵 ≤ 𝐴 𝐵
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4. Continue….
In general, it is not true that
𝐴𝐵 = 𝐴 𝐵 .
For instance consider
𝐴: 𝑙2 → 𝑙2: (𝑥1, 𝑥2, … ) → (𝑥1, 𝑥2, 0, … )
Then 𝐴 = 1 𝑏𝑢𝑡 𝐴2 = 0.
𝐴𝐴 = 𝐴2 = 0 = 0
𝐴 𝐴 = 1
𝐴𝐴 ≠ 𝐴 𝐴 .
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