7. Let T, and T, be bounded linear operators on a complex Hilbert space H into itself. If (T,x, x)=(T2x, x) for all xe H, show that T=T2. %3D

Q: 2. Define the map T:P2¬P4 by T(f) (x)=f(x²), Please determine the rank of I by using standard basis.

A:

Q: (d) If S,T are symmetric linear operators on an inner product space V, then their composition ST is…

A:

Q: . Show that the operators T₁,, T4 from R2 into R2 defined by (1, 2) (1, 0) (§1, §2) (0, §₂) (§1, §2)…

A:

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

A:

Q: Let X be a normed linear space and T: X→ X and S: X → X be two bounded linear operators, then SoT…

A:

Q: 15. If T is a self-adjoint linear operator on a finite-dimensional inner product space V, then detT…

A:

Q: 3. Let L be a linear operator on R and let L(1) = a . Show that L(x) = ax for all x E R .

A:

Q: Accept that (C'[0, 1], || - ||1) is a Banach space, where c'(0, 1] := {x € C[0, 1] : a'(t) E C[0,…

A: Consider the given C10,1:x∈C0,1:x'(t)∈C0,1 and x1=supt∈[0,1]x(t)+supt∈[0,1]x'(t)ddt=1

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

A:

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

A:

Q: Define an operator T: fºº → f° by T(5₁) = (¹) # and bounded. for each (₁) E. Show that T is linear…

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: Consider the linear operator defined by Show that T is compact. E1 E2 E3 T:l² l², x = (1, 2, 3, ...)…

A:

Q: 9 T1, T2 are normal operators on an laner produce space with the property that either commutes with…

A:

Q: Let L:R R be a linear operator such that L X2 X3- X2 then Ker(L) spanned by: Select one: (e, e )

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: 1) Find the norm of the following operators d) T:C[0,1] → C[0,1] by (Tx)(t) = f¿(t - s)*x(s)ds

A: Norm of operator gives the idea of size of any linear operator. Norm of can be found as .

Q: Show that if A is invertible and DA = CA, then D = C.

A: Given that, A is invertible and DA=CA.

Q: 1. Suppose V is a vector space over C and D: V x V → C is a sesqui-linear form. Show that is…

A: We will use the basic knowledge of linear algebra, complex numbers, and sesqui-linear forms to…

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: [#1] Exercise 7.46. Prove that if y = y' then y = ky' for some scalar k.

A: Given that γ≡γ' The y can be written as,  γ≡r mod n The y' can be written as, γ'≡ r mod n

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

A:

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

A:

Q: #Let Vz arih Here, れgl。 and hahe what netationship must hold betaen adho foro v to be ewoatly…

A: The expression for the volume is V=πr2h.   Thus, the total differential is given by dV=∂V∂rdr…

Q: Let L: R2 → R2² be the linear operator defined by L 2

A:

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

A:

Q: 2. Let T: R³ R³ be the operator given by 1 1 T(v) = 9). 1 Determine whether T is decomposable or…

A: Consider the given operator, Tv=-1100-1100-1v Here we need to determine that whether T is…

Q: Suppose A1,... ,An is a list of distinct real numbers. Prove that the list e^1x. ..., e^n* is…

A: According to the given information,

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: .

A: Given: L is the linear operator on and it is define by,

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

A:

Q: Let T be the linear thansformation from R³ to R4 defined by 1 -8 1 X1 1 3 2 X2 X2 -2 2 4 X3 X3 3 0…

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: 2) Find the indicated norms of the following linear functionals defined on RS b) f(x1,X2, X3, X4,X5)…

A: This question is about application of functional analysis

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: Prove that if A is invertible and BA = CA, then B = C.

A:

Q: Which of the following is true about the subset T = {1+ 2x - x², 2 + 3x + x²,4 + 7x − x²} of the…

A:

Q: 4) Let V be the vector space of polynomials of degree < 2 and let p(x) = ax² + bx + c. Find the…

A: Given that, The definition for adjoint linear operator is, Let V and U be two vector space over the…

Q: Assume that T : C° → C is a linear operator with minimal polynomial m7 (x) = (x – 2)2 . Find all…

A:

Q: Conversely, if T :V V is a linear operator such that T2 = T, show that V = kerT O imT. [Hint: 3.b, v…

A:

Q: Let TE B(H) be a self-adjoint operator on a Hilbert space H (a) Use the continuous functional…

A: Given that: T∈BH be a self-adjoint operator on a Hilbert space H. The objective is to show that: a…

Q: 6. Find a basis for the null space of the functional f defined on R³ by ƒ(x) = §1 +§2 − §3, where x…

A:

Q: Show that each of the following are linear operatorson R2. Describe geometrically what each…

A: According to the given information, it is required to show that it is the given is linear operator.

Q: Let T: P2 that is, 7. > P2 be the linear operator T(p(x)) = p(2x+ 1), T(co+ C,x+ C,x²) = c, + c,(2x…

A:

Q: Let T be a self-adjoint operator on a finite-dimensional inner product space V. Prove that for all x…

A:

Q: Let T be a linear operator on an inner product space V, and suppose that ||T(x)||= ||x|| for all x.…

A: According to property of norm the value of ||x||=0 iff x=0.

Q: Determine whether the following are linear operatorson Rn×n. L (A) = A − AT

A:

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.