Open mapping theorem, closed graph theorem, properties of conjugate operators - One Line Questions
1.
What is the conjugate of the identity operator I on a Hilbert space H? —
I
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: —
(T* - λ̄I)
3.
Let T and S be bounded linear operators on a Hilbert space H. Which property holds for their conjugate operators? —
All of the above
4.
Which of the following is NOT a property of conjugate operators on Hilbert spaces? —
(TS)* = S*T*
5.
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*? —
||T|| = ||T*||
6.
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? —
||T|| = 1
7.
The statement of the Closed Graph Theorem requires that the domain space is: —
A complete normed vector space (Banach space)
8.
The Open Mapping Theorem essentially states that a surjective, continuous linear map between Banach spaces is: —
An open map
9.
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? —
Mapping convergent sequences to convergent sequences.
10.
Let T: X → Y be a linear operator between Banach spaces. If T is bounded, what can be said about its graph G(T)? —
G(T) is a closed subspace if and only if T is continuous.
11.
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? —
The map x ↦ <Tx, y> must be a bounded linear functional for every y.
12.
What is the domain of the conjugate operator T* of a bounded linear operator T: H₁ → H₂, where H₁ and H₂ are Hilbert spaces? —
H₂
13.
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)? —
Open Mapping Theorem
14.
Which theorem is crucial in proving that if a linear operator T between Banach spaces is bounded below, then it is an open mapping? —
Open Mapping Theorem
15.
Which theorem is often used in conjunction with the Open Mapping Theorem to prove the Closed Graph Theorem? —
Uniform Boundedness Principle
16.
The Closed Graph Theorem is equivalent to which of the following statements for a linear operator T between two Banach spaces? —
If the graph of T is closed, then T is continuous.
17.
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? —
It is always bounded.
18.
If T is a linear operator on a Hilbert space H, and T = T*, then T is called: —
Self-adjoint
19.
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? —
Complete normed vector spaces (Banach spaces)
20.
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: —
Open sets to open sets.
21.
If T: X → Y is a linear operator between Banach spaces, and T is continuous, the Closed Graph Theorem implies that T is: —
Bounded
22.
Let T: H → H be a bounded linear operator on a Hilbert space. If T is normal, then T and T* commute. This implies: —
T commutes with T*.
23.
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: —
T is bounded.
24.
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? —
T is continuous.
25.
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? —
T is continuous.
26.
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? —
T is an open mapping.
27.
Consider T: X → Y, a linear operator between Banach spaces. If T is continuous and surjective, the Open Mapping Theorem guarantees that: —
T⁻¹ is continuous.
28.
Let T: X → Y be a linear operator between Banach spaces. If T is an open mapping, what does this imply about T? —
T is necessarily bounded.
29.
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? —
T is necessarily bounded.
30.
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? —
T is necessarily continuous.
31.
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? —
T is an open map.
32.
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? —
T is bounded.
33.
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? —
T is bounded.
34.
What is the primary condition required for the Open Mapping Theorem to apply to a linear operator T between two normed vector spaces? —
Both X and Y must be Banach spaces.
35.
For an operator T on a Hilbert space H, if T is self-adjoint, what property does it satisfy regarding its conjugate operator T*? —
T* = T
36.
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*? —
No specific relationship beyond definition.
37.
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? —
T* is always linear and bounded.
38.
If T is a bounded linear operator on a Hilbert space H, which of the following is true about its conjugate T*? —
T* is always bounded.
39.
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? —
T⁻¹ is continuous.
40.
If T: X → Y is a linear operator between Banach spaces, and T is bijective and continuous, what can be said about T⁻¹? —
T⁻¹ is continuous.
41.
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? —
The domain and codomain must be complete metric spaces (Banach spaces).
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)? —
The graph G(T) is a closed subspace if and only if T is continuous.
43.
What is the conjugate of the zero operator 0 on a Hilbert space H? —
The zero operator
44.
What is the conjugate (or adjoint) of an operator T on a Hilbert space H? —
The operator T* such that <Tx, y> = <x, T*y> for all x, y in H.
45.
For a self-adjoint operator T on a Hilbert space, T = T*. What is true about the spectrum of T? —
The spectrum is always contained in the real line.
46.
The Open Mapping Theorem is a powerful tool in functional analysis. Which of the following is a direct consequence of the Open Mapping Theorem? —
If a linear operator between Banach spaces is bounded below, it is an open map.
47.
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*? —
The zero operator
48.
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)? —
They are always real.
49.
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? —
They are always purely imaginary or zero.