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

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: A necessary and sufficient condition that a linear transformation P on a complex inner product space…

A:

Q: Prove the spectral theorem for Hermitian operators, H, in a finite dimensional inner product space…

A: Spectral theorem for Hermitian operators H in a finite dimensional inner product space V:…

Q: Define A: C → C° by A(z,, Z2, Z3) = (a,z1, azz2, lzz), «; e C. When is A a positive operator ?

A: A is a positive operator if all the eigen values are non negative Therefore, T(1, 0, 0) = (α1, 0,…

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: Describe the zero vector (the additive identity) of the vector space. P3 x + x²+

A: Every vector space has an additive identity. Additive identity is the same for all vectors in the…

Q: (12) Let T be a diagonalizable linear operator on V-finite dimensional, and let m be any positive…

A:

Q: be a linear operator on a vector space V(F). If = 0, t ce oft and null space of t. Also given an…

A:

Q: Let V be a finite-dimensional vector space and let T be a linear operator on V . Suppose that…

A:

Q: Why do the set of all polynomials of degree n or less form a vector space, while the set of all…

A: Why do the set of all polynomials of degree n or less form a vector space, while the set of all…

Q: then L is a linear operatorE Let L(x): R R be a mapping such that L Select one: True False

A:

Q: Prove that the adjoint operator is linear and bounded

A:

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

A: It is given that T is a unitary operator on a finite dimensional inner product space.

Q: 3.) Let V be a finite-dimensional vector space. Suppose T : V → V is a linear transformation. how…

A:

Q: Show that the element 0 in a vector space is unique

A:

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

A:

Q: then L is a linear operator. Let L(x). R R be a mapping such that L( X2 Select one: O True O False

A: We will check whether the given statement is true or false.

Q: (c) Show that T-l is not bounded, and explain why this is not in contradiction with the Open Mapping…

A: Af first we will show that T is bijective and hence its inverse T -1 exists. Then we will find that…

Q: Let T be an linear operator on a complex vector

A: let J=2100000002100000002000000002000000003100000003000000000100000000=[T]B Matrix J is in Jordan…

Q: Exercise 8. Show that a function F: R" → R" preserves the norm || · || if and only if it preserves…

A: We have the function F:ℝn→ℝn that preserves the norm · This implies that this function satisfies all…

Q: Prove that the composition of unitary [orthogonal] operators is unitary [orthogonal].

A: Let consider U  and V are the unitary operators.

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

A:

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 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: 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: 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 L(x): R² → R be a mapping such that L X2 CH3 X2 then L is a linear operator. Select one: O True…

A: The given linear mapping is: L(x): ℝ2→ℝ2    such that:  Lx1x2=x2x1

Q: Let A be an operator on a Hilbert space H. Show that A is normal if and only if ||Ax|| =||A*x|| for…

A: Suppose A is a normal operator. Then by definition,  AA*=A*A            1 Now, consider  Ax2=Ax, Ax…

Q: (b) Show that the set of all positive real numbers, with x+y and cx redefined to equal the usual xy…

A: Here, to check that the set of all positive real numbers with  x+y=xy and cx=xc where xy is usual…

Q: Theorem 3. Let A be a bounded linear operator acting between two real Hilbert spaces. Then 1.…

A: Given: A is a bounded linear operator acting between real Hilbert spaces. To prove: 1) R(A)⊥=NA* 2)…

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: im(T1 + 14T) im(14T† + T2).

A:

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

A:

Q: Prove or disprove whether the function f (r) = a +2 is a bijection from R R %3D

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: Prove that the function g:R → R* • defined by g(x) = 2* is an injective homomorphism that is not…

A:

Q: Let V be an inner product space and T be a linear operator on V with || Tv||= ||v||  ∀ v ∈ V. Show…

A: Let V be an inner product space and T be a linear operator on V with Tv=v ∀v∈V.

Q: Let f map C2 into C2 be a linear operator such that for every x in C2, f(f(x)) = x. Let V+ = {x in…

A: Let f:C2→C2 be a linear operator such that for every x∈C2, ffx=x.  Suppose V+=x∈C2:fx=x and…

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

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: How can you prove that T ∈ L(FF) are isomorphisms when considering a vector space (FF)={(x1,x2)|when…

A:

Q: Let A be an operator on a Hilbert space H. Show that A is normal if and only if ||Ax|| = ||A*x|| for…

A:

Q: Let T be a linear operator on a finite-dimensional inner product space V. (a) If T is an orthogonal…

A: Let T be a linear operator on a finite-dimensional inner product space V.(a)Let T be an orthogonal…

Q: A real-valued function f defined on the real line is called an even function if  J( -t) = f (t) for…

A: Givenf(-t)=f(t)-even functions.To prove Set of even functions forms a vector space.

Q: A function is invertible if and only if it is a bijection. True or False?

A:

Q: Theorem 3. Let A be a bounded linear operator acting between two real Hilbert spaces. Then 1.…

A:

Q: Which of the following vector spaces is not isomorphic to all the others R6  R6[x]  M2×3(R)…

A: