2. Let H H H where H be a Hilbert space. Let A = B(H) and B be the operator defined on H by 0 iA B = -iA* 0 Prove that || A|| = ||B|| and that B is self-adjoint.

Q: Consider the right-shift on the Hilbert space l2: S:l → l, S(@1,a2, ...) = (0, a1, a2, ...). Define…

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Q: _I. Let H and K be Hilbert spaces and A E B(H, K). Show that A is compact if = and only if A* A is…

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Q: Prove that the adjoint operator is linear and bounded

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Q: values of (1, Ar) on the manifold where (1, B1): Let A and B be Hermitian operators on a real…

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Q: (c) Prove that the adjoint of a compact operator is compact.

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Q: Consider polynomials with degree less than or equal to n. We can think that each polynomial…

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Q: Q6: Show that R³ = {(x1,x2, x3)|lx1,x2, x3 E R} is a vector space

A: Vector space is the field created by set of vectors with specified direction and magnitude, usually…

Q: 1. If two Hermitian operators A and B commute, show that they can be simultaneously diagonalized.

A: Given :-  If two Hermitian operators A and B commute , then they can be simultaneously diagonalized.

Q: Let T be a self-adjoint operator on a Hilbert space H. Show that its norm is given by ||T|| = sup…

A: Given that T is a self-adjoint operator on a Hilbert space H. To prove: T=supx=1Tx,x.

Q: Let S be the set of all ordered pairs of real numbers (x1, x2) equipped with the following…

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Q: Let V be a finite dimensional vector space, and T:        V------->V be a linear operator. Let NT…

A: Let dimension of vector space V is n. Case (1) Let T be an invertible operator this implies that T…

Q: Show that a bounded linear operator A on complex Hilbert space H is self adjoint if and only if is…

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Q: Let A = X = 1 −1 1 1 −1 1 1 -1 1 -1 −1 1 1 1 -1 - Find a non-zero vector x in the null space of A.

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Q: Let T be abounded linear operator on a Hilbert space H. Then T, =(T +T) is Select one: O none O (T'…

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Q: Let T be a diagonalizable linear operator on   V -finite-dimensional, and let m be any positive…

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Q: Let H be a Hilbert space and A be a positive operator on H. Prove that I + A is invertible in BL…

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Q: im(T1 + 14T) im(14T† + T2).

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Q: Let T be a linear operator on a finite-dimenstona! inner product space V. Let E1,..., Ex be linear…

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Q: Let V be the set of all pairs (x,y) of real numbers together with the following operations:…

A: Let  be the set of all pairs  of real numbers together with the following operations:

Q: Let T1and T2 be normal

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Q: Let A E BL(H). Then A is called normal if A*A = AA*, uni- tary if A*A = I = AA*, that is, A* = A−¹,…

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Q: 1. Let V be the set of all ardered pairs of real numbers, and consider the following addition and…

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Q: Let A be a bounded linear operator on a Hilbert space H and ||A(x)||=||x|| ¥xe H. Show that A is…

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Q: Determine whether or not each of the following properties apply to the set of polynomials {1-x+2x²,…

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Q: (5) Let Pe B(H) be a nonzero idempotent. Show that the follow- ing are equivalent. (a) P is a…

A: It is provided that P∈BH is a nonzero idempotent.  We have to the following are equivalent: (a) P is…

Q: Find the transpose, conjugate, and adjoint of 6-3i 2+12i -19i 0 5+2.1i 17 1 2+5i 3-4.5i

A: A=6-3i2+12i-19i05+2.1i1712+5i3-4.5i