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

Q: Show that A is invertible if and only if AT is invertible.

A: Concept: Where the operator, operating on a function, produces a constant times the function, is…

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

A:

Q: Prove that if T is a unitary operator on a finite-dimensional inner product space V, then T has a…

A: Since, T is a unitary operator, there exists an orthonormal basis consisting of eigen vectors of T…

Q: 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 =…

A: As per the question we are given a matrix operator B on the Hilbert space H. And we have to show :…

Q: Let P3 be the vector space of polynomials of degree at most 3, and let S = {p1(x), p2(x), p3(x), p…

A:

Q: Give an example of a linear operator T on R2 and an ordered basis for R2 that provides a…

A:

Q: If H is a Hilbert space and A € BL (H) is positive, prove that A+21 is invertible.

A:

Q: Let T be a linear operator on a finite dimensional vector space, V and suppose that the different…

A:

Q: Let A be an invertible matrix and let Az = b (A+AA)(x+ Ar) = b. inequality ||AT|| A

A:

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

A:

Q: If A is a bounded linear operator A on a Hilbert space H satisfies the condition that ||Ax|| = ||A*x…

A: According to the given information, it is required to solve the part (c). Consider the given…

Q: ITA Is a bounded self-adjoint operator on a Hilbert space н, show that |= sup {|}: xe H |x| = 1}

A:

Q: Consider the linear operator L: BC[0, ] → BC[0, 7] defined by Си — и" - 2и is unbounded in the sup…

A: Consider the linear operator L:BC0,π→BC0,π defined by: Lu=u''-2u. We know that the set BC0,π is…

Q: Let T be an invertible linearlo-perator on a finite-dimensionalvector spaceProve that T is…

A:

Q: Describe all linear operators T on R2 such that T is diagonalizable and T3 - 272 +T= To.

A:

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 H, and H, be Hilbert spaces and T: H H, a bounded linear operator. If M,cH, and M2 c H2 are such…

A: Given H1 and H2 be Hilbert space and T:H1→H2 a bounded linear operator. If M1⊂H1 and M2⊂H2 are such…

Q: Let U be a unitary matrix. Prove that ||Ux|| = ||x|| for all x ∈ Cn.

A:

Q: Let H be a Hilbert space and T: H Ha bijective bounded linear operator whose inverse is bounded.…

A: Given H be the Hilbert space and T:H→H a bijective bounded linear operator whose inverse is bounded.…

Q: Let L be the linear operator on R³ defined by 2x1 – x2 – X3 L(x) = | 2x2 – x1 – X3 2x3 – X1 – X2…

A:

Q: Let H1 and H2 be any two Hilbert Spaces and T : H1 → H2 be a bounded linear operator. Let M1 = N(T)…

A: Given:- Let H1 and H2 be any two Hilbert Spaces and T : H1 → H2 be a boundedlinear operator. Let M1…

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

A:

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

A:

Q: Let A€BL(H) be a compact operator, then show that A*A is a compact

A:

Q: Let T be a linear operator on a vector space V, and let γ be a cycle of generalized eigenvectors…

A: Let T  be a linear operator on a vector space V.

Q: Show that if a non-zero element in a normed space X, then there is a bounded linear functional f on…

A: We need to show that if a non-zero element in a normed space X, then there is a bounded linear…

Q: LetT be a linear operator on an inner product space V. Let U1 = T + T*and U2 = TT*. Prove that U1 =…

A: T is an operator in an inner product space V over the field F together with an inner product. V×V→F…

Q: Let A be an n × n matrix and let λ be a scalar. Show that det(A − λI) = 0 if and only if Ax = λx for…

A:

Q: 3. Let A, B, T be bounded self-adjoint linear operators on a complex Hilbert space H. If T20 and…

A:

Q: State and prove the : The characteristic polynomial of any diagonalizable linear operator T splits.

A: We have to prove that: The characteristic polynomial of any diagonalizable linear operator T splits.

Q: Let P3 be the vector space of polynomials of degree at most 3, and let S = {p1(x), p2(x), p3(x), p…

A:

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

A:

Q: et L:RR be a linear operator such that L x2 )-X1, then Im(L) spanned by= X3 X1 Gelect one: a. O b.…

A: T : A  → B thenIm(T) = { TX : X ∈ A}

Q: Describe all linear operators T on R2 such that T is diagonalizable and T3−2T2 + T = T0.

A:

Q: Let X be a pre-Hilbert space then [(x, y)|

A:

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

A:

Q: Let T: H- » H be a bounded positive self-adjoint linear operator on a complex Hilbert space. Using…

A: As per the question we have to show that the Cauchy-Schwarz inequality holds for the Hilbert space H…

Q: Let f1(x) = 3x and f2(x) = ∣x∣. Graph both functions on the interval −2 ≤ x ≤ 2. Show that these…

A:

Q: Let X be a Hilbert space and T:X → X and T2: X → X are self-adjoint operators such that either…

A: Given we have given that X is a Hilbert space and T1 : X→X & T2 :X→X  are self -adjoint…

Q: Let H be a Hilbert space and A E B(H). Show that (ker A*)+. (a) Image A = (b) ker A = (Image A*)+.

A:

Q: prove that A is invertible and A = B = C. %3D

A:

Q: Let X be a normed linear space such that its dual X' is separable, then show that X is separable ?…

A: This is a problem of Functional Analysis.

Q: Let V be a finite dimensional inner product space and T a non negative operator on V. Then T has a…

A:

Q: Let L be a linear operator on a vector space V. Define Ln, n ≥ 1, recursively by L1 = L Lk+1(v) = L…

A:

Q: Define A : L3/2[–1, 1] → C by Prove that A is a bounded linear map and compute its norm.

A:

Q: Let T be a linear operator on a finite-dimensional Prove that T is normal if and only if --- omplex…

A:

Q: Let T : V → V be a linear operator on a finite-dimensional complex vector space. A cycle of…

A: This question is about application of linear algebra

Q: Q3- Let S and T be bounded linear operators on a complex Hilbert space H into itself. If (Sx, x) =…

A: Let S and T be bounded linear operator on a complex Hilbert space H into itself. If…