(d) Show that T is Hermitian if and only if (Tx,x) E R for all x EH. (Hint: Note
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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![Problem 2. Let H be an Hilbert space. For all operator T on H, we define
v(T) = sup{(Tx, x); ||x|| = 1}.
1. (a) Show that v is a semi-norm on the space L(H) of operators on H.
(b) Show that (Tx, x)| ≤ v(T)||x||², for all x € H.
(c) Show that T is Hermitian and that if v(T) = 0 then T = 0.
2. Suppose that H is a complex Hilbert space not reduced to {0}.
(a) Let U be a Hermitian operator on H and let x, y € H.
i. For all t€ R, Expand (U(x+ty), x+ty) — (U(x-ty), x-ty) then deduce
that
4|t||Re((Ux, y))| ≤ v(U)(||x + ty||² + ||x − ty||²).
|Re((Ux, y))| ≤ v(U)||x||||x||.
ii. Deduce that
(b) Show that, for all operator S: H→ H, we have
Re[(Sx, y) + (x, Sy)]| < 2v(S) ||x||||x||.
(c) Deduce that, for all operator T on H, we have
v(T) ≤ ||T|| ≤ 2v(T).
(Hint: For all x
H, choose a complex number λ of modulus 1 such that
X²(T²x, x) € R+, then take S = XT and y = Sx).
(d) Show that T is Hermitian if and only if (Tx,x) ≤ R for all x € H. (Hint: Note](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9612a6a5-e19c-4346-ae6e-6941d4a9f0e7%2Fbf947532-a063-4cbd-971b-2e26b1205f19%2Fcrng4nf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 2. Let H be an Hilbert space. For all operator T on H, we define
v(T) = sup{(Tx, x); ||x|| = 1}.
1. (a) Show that v is a semi-norm on the space L(H) of operators on H.
(b) Show that (Tx, x)| ≤ v(T)||x||², for all x € H.
(c) Show that T is Hermitian and that if v(T) = 0 then T = 0.
2. Suppose that H is a complex Hilbert space not reduced to {0}.
(a) Let U be a Hermitian operator on H and let x, y € H.
i. For all t€ R, Expand (U(x+ty), x+ty) — (U(x-ty), x-ty) then deduce
that
4|t||Re((Ux, y))| ≤ v(U)(||x + ty||² + ||x − ty||²).
|Re((Ux, y))| ≤ v(U)||x||||x||.
ii. Deduce that
(b) Show that, for all operator S: H→ H, we have
Re[(Sx, y) + (x, Sy)]| < 2v(S) ||x||||x||.
(c) Deduce that, for all operator T on H, we have
v(T) ≤ ||T|| ≤ 2v(T).
(Hint: For all x
H, choose a complex number λ of modulus 1 such that
X²(T²x, x) € R+, then take S = XT and y = Sx).
(d) Show that T is Hermitian if and only if (Tx,x) ≤ R for all x € H. (Hint: Note
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