(b) Show that, for all operator S: H→ H, we have Re[(Sx, y) + (x, Sy)]| ≤ 2v(S)||1|||1||. (e) Deduce that, for all operator T on H, we have v(T) < ||T|| < 2v(T). (Hint: For all x H, choose a complex number of modulus 1 such that X² (T²r, x) € R+, then take S = XT and y = Sx). (d) Show that T is Hermitian if and only if (T1, 1) € R for all z € H. (Hint: Note that TT if and only if v(T-T) = 0).
(b) Show that, for all operator S: H→ H, we have Re[(Sx, y) + (x, Sy)]| ≤ 2v(S)||1|||1||. (e) Deduce that, for all operator T on H, we have v(T) < ||T|| < 2v(T). (Hint: For all x H, choose a complex number of modulus 1 such that X² (T²r, x) € R+, then take S = XT and y = Sx). (d) Show that T is Hermitian if and only if (T1, 1) € R for all z € H. (Hint: Note that TT if and only if v(T-T) = 0).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
hilber space hermitian part 2 b c d e
![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 z € 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||||T||.
ii. Deduce that
(b) Show that, for all operator S: H → H, we have
Re[(Sx, y) + (x, Sy)]| ≤ 2v(S)||1||||1||.
(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 A 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
that TT if and only if v(T-T*) = 0).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b5ce474-09f2-43c8-b918-66fe96c70f92%2F01b0b40c-e5eb-4764-bd47-cbd2c70c2ee9%2Fcxo6eg8_processed.png&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 z € 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||||T||.
ii. Deduce that
(b) Show that, for all operator S: H → H, we have
Re[(Sx, y) + (x, Sy)]| ≤ 2v(S)||1||||1||.
(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 A 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
that TT if and only if v(T-T*) = 0).
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