Consider the Hilbert space H = Let (X, , μ) be a o-finite measure space. L²(X, N, μ) =: L² (µ). If ¢ € L∞ (μµ), define M : L² (μ) → L² (μ) by Mof=of. (a) Show that Mo B(H) and ||Mo|| = ||||. Here is the u-essential supremum norm. Mō. (b) Show that Mr = (c) Show that Mo is normal. When Mo is self adjoint? unitary?

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Consider the Hilbert space H =
Let (X, ,μ) be a o-finite measure space.
L²(X, N, µ) =: L² (µ). If ¢ € L∞ (μµ), define
M : L² (μ) → L² (μ) by Mof=of.
(a) Show that
M = B(H) and ||M|| = ||||.
Here
is the u-essential supremum norm.
Mō.
(b) Show that Mr
=
(c) Show that Mo is normal. When Mo is self adjoint? unitary?
Transcribed Image Text:Consider the Hilbert space H = Let (X, ,μ) be a o-finite measure space. L²(X, N, µ) =: L² (µ). If ¢ € L∞ (μµ), define M : L² (μ) → L² (μ) by Mof=of. (a) Show that M = B(H) and ||M|| = ||||. Here is the u-essential supremum norm. Mō. (b) Show that Mr = (c) Show that Mo is normal. When Mo is self adjoint? unitary?
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