TRB PG Assistant - Open mapping theorem, closed graph theorem, properties of conjugate operators Question Bank with Options

1. The Closed Graph Theorem states that if T: X → Y is a linear operator between Banach spaces, and its graph is closed, then T is bounded. This implies that if T is continuous, then:

A) T is an open map.

B) T is surjective.

C) T is bounded.

D) T⁻¹ is continuous.

2. Let T: H → H be a bounded linear operator on a Hilbert space. If T is self-adjoint (T = T*), then for any scalar λ, the operator (T - λI) has a conjugate (T - λI)* equal to:

A) (T* - λ̄I)

B) (T* - λI)

C) (T - λ̄I)

D) (T* + λI)

3. If T: X → Y is a linear operator between Banach spaces, and T is bijective and continuous, what can be said about T⁻¹?

A) T⁻¹ is unbounded.

B) T⁻¹ is not defined.

C) T⁻¹ is continuous.

D) T⁻¹ is not necessarily linear.

4. The Open Mapping Theorem is a consequence of the Baire Category Theorem applied to the space of operators. What is a key requirement for its application?

A) The domain and codomain must be Hilbert spaces.

B) The domain and codomain must be complete metric spaces (Banach spaces).

C) The operator must be compact.

D) The operator must be injective.

5. Let T: H → H be a bounded linear operator on a Hilbert space. If T is normal, then T and T* commute. This implies:

A) T is always self-adjoint.

B) T is always unitary.

C) T* commutes with T.

D) T commutes with T*.

6. Consider a linear operator T: X → Y between Banach spaces X and Y. If the graph of T is closed, what does the Closed Graph Theorem imply about T?

A) T is an open mapping.

B) T is continuous.

C) T is surjective.

D) T is injective.

7. What is the conjugate of the zero operator 0 on a Hilbert space H?

A) The identity operator

B) The zero operator

C) Undefined

D) The adjoint operator

8. If T: X → Y is a linear operator between Banach spaces, and T is continuous, the Closed Graph Theorem implies that T is:

A) Surjective

B) Injective

C) Bounded

D) Compact

9. The Open Mapping Theorem essentially states that a surjective, continuous linear map between Banach spaces is:

A) An isometry

B) An open map

C) A compact map

D) A projection

10. Let T: H → H be a bounded linear operator on a Hilbert space. The conjugate operator T* has the property that <Tx, y> = <x, T*y>. Which of the following is true?

A) T* is always linear and unbounded.

B) T* is always linear and bounded.

C) T* is always non-linear and bounded.

D) T* is always non-linear and unbounded.

11. For a self-adjoint operator T on a Hilbert space, T = T*. What is true about the spectrum of T?

A) The spectrum is always empty.

B) The spectrum is always contained in the imaginary axis.

C) The spectrum is always contained in the real line.

D) The spectrum is always contained in the unit circle.

12. If T: X → Y is a linear operator between Banach spaces, and T is bounded, what can be said about the image of a bounded set under T?

A) It is always bounded.

B) It is always unbounded.

C) It is not necessarily bounded.

D) It is always compact.

13. Let T: X → Y be a linear operator between Banach spaces X and Y. If T is surjective and continuous, the Open Mapping Theorem guarantees that T maps:

A) Open sets to open sets.

B) Closed sets to closed sets.

C) Bounded sets to bounded sets.

D) Compact sets to compact sets.

14. Which theorem is often used in conjunction with the Open Mapping Theorem to prove the Closed Graph Theorem?

A) Hahn-Banach Theorem

B) Uniform Boundedness Principle

C) Cauchy Integral Formula

D) Spectral Theorem

15. The conjugate operator T* of a bounded linear operator T: H₁ → H₂ is defined such that <Tx, y> = <x, T*y>. If T is the zero operator, what is T*?

A) The zero operator

B) The identity operator

C) Undefined

D) The inverse of T

16. If T: H → H is a bounded linear operator on a Hilbert space H, and T is normal (TT* = T*T), what is the relationship between T and T*?

A) T* = T

B) T* = -T

C) T* = T⁻¹

D) No specific relationship beyond definition.

17. Let T: X → Y be a linear operator between Banach spaces. If T is bounded, what can be said about its graph G(T)?

A) G(T) is always an open set in X × Y.

B) G(T) is always a closed set in X × Y.

C) G(T) is a closed subspace if and only if T is continuous.

D) G(T) is a closed subspace if and only if X is finite-dimensional.

18. The Open Mapping Theorem is a key result in establishing the equivalence between boundedness and other properties of linear operators. Which of these properties is NOT equivalent to boundedness for a linear operator between Banach spaces?

A) Being an open map (if surjective).

B) Having a continuous inverse (if bijective).

C) Mapping Cauchy sequences to Cauchy sequences.

D) Mapping convergent sequences to convergent sequences.

19. Which of the following is NOT a property of conjugate operators on Hilbert spaces?

A) (T+S)* = T* + S*

B) (αT)* = ᾱT*

C) (TS)* = S*T*

D) (T*)* = T

20. If T: X → Y is a linear operator between Banach spaces X and Y, and T is continuous, which of the following is guaranteed by the Closed Graph Theorem?

A) T is surjective.

B) T is an open map.

C) T is bounded.

D) T⁻¹ exists and is continuous.

21. What is the conjugate of the identity operator I on a Hilbert space H?

A) -I

B) I⁻¹

C) I

D) 0

22. Let T: X → Y be a linear operator between Banach spaces. If T is an open mapping, what does this imply about T?

A) T is necessarily continuous.

B) T is necessarily bounded.

C) T maps bounded sets to bounded sets.

D) T maps closed sets to closed sets.

23. The statement of the Closed Graph Theorem requires that the domain space is:

A) A Hilbert space

B) A complete normed vector space (Banach space)

C) A finite-dimensional vector space

D) A separable Hilbert space

24. If T: H → H is a bounded linear operator on a Hilbert space, and T* = -T (skew-adjoint), what can be said about the eigenvalues of T?

A) They are always real and non-zero.

B) They are always purely imaginary or zero.

C) They are always positive.

D) They are always negative.

25. Let T be a linear operator on a Hilbert space H. If T is unitary, meaning T*T = TT* = I, what can be said about its norm?

A) ||T|| < 1

B) ||T|| > 1

C) ||T|| = 1

D) ||T|| is undefined.

26. If T: X → Y is a linear operator between Banach spaces, and T is injective and has a closed graph, what can be concluded about T?

A) T is necessarily unbounded.

B) T is necessarily an open map.

C) T is necessarily continuous.

D) T is necessarily surjective.

27. Consider T: X → Y, a linear operator between Banach spaces. If T is continuous and surjective, the Open Mapping Theorem guarantees that:

A) T is injective.

B) T⁻¹ is continuous.

C) T maps bounded sets to bounded sets.

D) T maps compact sets to compact sets.

28. The Open Mapping Theorem is a powerful tool in functional analysis. Which of the following is a direct consequence of the Open Mapping Theorem?

A) The Uniform Boundedness Principle.

B) If a linear operator between Banach spaces is bounded below, it is an open map.

C) The existence of non-trivial bounded linear functionals.

D) The Riesz Representation Theorem.

29. If T is a bounded linear operator on a Hilbert space H, what can be said about the eigenvalues of a self-adjoint operator (T* = T)?

A) They are always complex.

B) They are always non-negative.

C) They are always real.

D) They are always distinct.

30. Let T and S be bounded linear operators on a Hilbert space H. Which property holds for their conjugate operators?

A) (T+S)* = T* + S*

B) (TS)* = T*S*

C) (T*)* = T

D) All of the above

31. For a linear operator T on a Hilbert space H, the conjugate operator T* is defined using the inner product. What is the condition for T* to exist for any bounded linear operator T?

A) H must be finite-dimensional.

B) H must be a Banach space.

C) The map x ↦ <Tx, y> must be a bounded linear functional for every y.

D) T must be surjective.

32. If T is a linear operator on a Hilbert space H, and T = T*, then T is called:

A) Normal

B) Unitary

C) Self-adjoint

D) Isometry

33. Let T: H₁ → H₂ be a bounded linear operator between Hilbert spaces. The conjugate operator T*: H₂ → H₁ satisfies <Tx, y> = <x, T*y> for all x ∈ H₁ and y ∈ H₂. What is the relationship between the norm of T and the norm of T*?

A) ||T|| < ||T*||

B) ||T|| > ||T*||

C) ||T|| = ||T*||

D) ||T|| ≤ ||T*||²

34. If T is a bounded linear operator on a Hilbert space H, which of the following is true about its conjugate T*?

A) T* is always unbounded.

B) T* is always compact.

C) T* is always bounded.

D) T* is never self-adjoint.

35. What is the domain of the conjugate operator T* of a bounded linear operator T: H₁ → H₂, where H₁ and H₂ are Hilbert spaces?

A) H₁

B) H₂

C) The set of all bounded linear functionals on H₂.

D) The set of all vectors y in H₂ such that the map x ↦ <Tx, y> is bounded.

36. For an operator T on a Hilbert space H, if T is self-adjoint, what property does it satisfy regarding its conjugate operator T*?

A) T* = -T

B) T* = T⁻¹

C) T* = T

D) T* = -T⁻¹

37. What is the conjugate (or adjoint) of an operator T on a Hilbert space H?

A) The inverse of T, T⁻¹.

B) The operator T* such that <Tx, y> = <x, T*y> for all x, y in H.

C) The operator T such that <Tx, y> = <y, T*x> for all x, y in H.

D) The operator T such that <Tx, y> = <T*x, y> for all x, y in H.

38. If X is a Banach space and Y is a normed vector space, and T: X → Y is a linear operator such that its graph is closed, what does the Closed Graph Theorem imply?

A) T is continuous.

B) T is surjective.

C) T is injective.

D) T is an open mapping.

39. Let T: X → Y be a linear operator between Banach spaces. If T is bijective and its inverse T⁻¹ is continuous, what can be inferred about T?

A) T is not necessarily an open map.

B) T is an open map.

C) T is not necessarily closed.

D) T maps closed sets to closed sets.

40. Consider a linear operator T: X → Y between Banach spaces X and Y. If T is continuous, what property does the Closed Graph Theorem guarantee for T?

A) T is surjective.

B) T maps open sets to open sets.

C) T is bounded.

D) The graph of T is open.

41. Which theorem is crucial in proving that if a linear operator T between Banach spaces is bounded below, then it is an open mapping?

A) Hahn-Banach Theorem

B) Cauchy-Schwarz Inequality

C) Open Mapping Theorem

D) Spectral Theorem

42. Let T: X → Y be a linear operator between Banach spaces X and Y. If T is bounded, what can be said about its graph G(T)?

A) The graph G(T) is always open in X × Y.

B) The graph G(T) is always closed in X × Y.

C) The graph G(T) is a closed subspace if and only if T is continuous.

D) The graph G(T) is a closed subspace if and only if T is surjective.

43. The Open Mapping Theorem is fundamentally about the relationship between the continuity of a linear operator and its ability to map open sets to open sets, provided the domain and codomain are what type of spaces?

A) Normed vector spaces

B) Complete normed vector spaces (Banach spaces)

C) Hilbert spaces

D) Finite-dimensional vector spaces

44. If T: X → Y is a linear operator between Banach spaces X and Y, and T is both injective and surjective (a bijection), what can be concluded if T is also continuous?

A) T⁻¹ is unbounded.

B) T⁻¹ is not defined.

C) T⁻¹ is continuous.

D) T is not necessarily an open map.

45. What is the primary condition required for the Open Mapping Theorem to apply to a linear operator T between two normed vector spaces?

A) T must be injective.

B) T must be surjective.

C) Both X and Y must be Banach spaces.

D) T must be compact.

46. Consider a linear operator T: X → Y, where X and Y are Banach spaces. If the graph of T, denoted by G(T) = {(x, Tx) : x ∈ X}, is a closed subspace of X × Y, what does the Closed Graph Theorem imply about T?

A) T is necessarily surjective.

B) T is necessarily bounded.

C) T is necessarily compact.

D) T is necessarily an isometry.

47. Let X and Y be Banach spaces and let T: X → Y be a linear, continuous operator. If T is surjective, what can be said about T?

A) T is injective.

B) T is an open mapping.

C) T is compact.

D) T is self-adjoint.

48. The Closed Graph Theorem is equivalent to which of the following statements for a linear operator T between two Banach spaces?

A) If the graph of T is closed, then T is continuous.

B) If T is continuous, then its graph is closed.

C) If T is surjective and continuous, then T is an open map.

D) If T maps open sets to open sets, then T is continuous.

49. Which theorem states that if T is a linear operator between two Banach spaces, and T is continuous (or bounded), then its inverse T⁻¹ is also continuous (or bounded)?

A) Hahn-Banach Theorem

B) Riesz Representation Theorem

C) Open Mapping Theorem

D) Uniform Boundedness Principle

Open mapping theorem, closed graph theorem, properties of conjugate operators - Question Bank