Question

Let \(H\) be a finite-dimensional Hilbert space. Follow the hint to show that \(\operatorname{Ext}\left(B_{\mathcal{B}(H)}\right)\) consist exactly of all unitary operators. The statement is in fact true for all Hilbert spaces. In the hint, we use several facts from the operator theory. First, for every \(T \in \mathcal{B}(H)\) there exists a polar decomposition \(T=U S\), where \(U\) is a unitary operator and \(S\) is a positive operator; that is, \((S(x), x) \geq 0\) for every \(x \in H\). Second, if \(H\) is a finite-dimensional Hilbert space and \(S\) is a positive operator on \(H\), we can find an orthogonal basis of \(H\) so that the matrix of \(S\) with respect to this basis is diagonal. Note that elements on the diagonal have magnitudes at most \(\|S\|\).

Short answer

The extreme points of bounded operators consist exactly of all unitary operators in a finite-dimensional Hilbert space.

Step-by-step solution

Understand Polar Decomposition

Recall that for every operator \(T\) in \(\mathcal{B}(H)\), there exists a polar decomposition \(T = US\), where \(U\) is a unitary operator and \(S\) is a positive operator. This decomposition helps us separate \(T\) into its 'rotational' (unitary) and 'scaling' (positive) components.

Define Extreme Points

Extreme points in a convex set are those that cannot be expressed as a convex combination of other points in the set. For the set of bounded operators \(\mathcal{B}(H)\), we need to identify operators that can't be decomposed this way unless they are unitary themselves.

Unitary Operators as Extreme Points

Unitary operators \(U\) satisfy \(U^*U = UU^* = I\). Because unitaries preserve inner product and norm, they are natural candidates for extreme points. If \(T\) can be expressed as \(T = \frac{1}{2}(T_1 + T_2)\), then \(U\) must also satisfy the unitary condition if \(T_1\) and \(T_2\) are units.

Explore the Positive Operator \(S\)

If \(S\) is a positive operator, it is diagonalizable with non-negative eigenvalues. This diagonal form suggests \(S\) scales vectors without changing their direction in specific subspaces associated with the eigenvectors.

Combining \(U\) and \(S\)

Since \(T = US\), any operator \(T\) can be uniquely identified by its unitary part \(U\) and its positive part \(S\). For \(T\) to remain an extreme point, \(S\) must appropriately 'scale' vectors while \(U\) maintains the structure of \(T\). Therefore, if \(S\) is the identity operator which doesn't scale, then \(T\) is purely unitary.

Conclusion

Given the above, we see that the extreme points \(\text{Ext}(B_{\mathcal{B}(H)})\) consist exactly of all unitary operators due to the unique nature of the polar decomposition and the properties of unitaries and positive operators in finite dimensions.

Key concepts

Polar Decomposition

In finite-dimensional Hilbert spaces, polar decomposition is a critical concept. It tells us that for any operator\( T \) in the set of bounded linear operators, denoted as\( \mathcal{B}(H) \, \), we can decompose it into two distinct parts: a unitary operator \( U \) and a positive operator \( S \). This is expressed as \( T = US \).

The unitary operator \( U \) represents the rotational component, which preserves the lengths of vectors. The positive operator \( S \) represents the scaling component, where each vector is scaled by a non-negative factor. Consider this due to the property of positive operators: \( (S(x), x) \geq 0 \) for every vector \( x \) in the Hilbert space.

With polar decomposition, complex transformations in finite-dimensional Hilbert spaces can be better understood by separately analyzing their rotational and scaling behaviors. This decomposition simplifies the problem and provides insights into the behavior of bounded operators.

Unitary Operators

Unitary operators in a Hilbert space are defined by their critical property: they preserve the inner product and, hence, the norm of vectors. Mathematically, an operator \( U \) is unitary if it satisfies \( U^* U = U U^* = I \), where \( U^* \) is the adjoint of \( U \) and \( I \) is the identity operator.

Unitary operators can be seen as 'perfect' rotations or reflections. They don't change vector magnitudes and are essential in various quantum mechanics, signal processing, and linear algebra problems.

This preservation of structure makes unitary operators vital in the study of extreme points in convex sets of bounded operators. Since they maintain the inner product, unitary operators often stand out as natural candidates for extreme points.

Positive Operators

Positive operators are those that return a non-negative inner product for any vector they act upon. Formally, an operator \( S \) is positive if\( (S(x), x) \geq 0 \) for every\( x \) in the Hilbert space \( H \).

One of the key properties of positive operators is that they are diagonalizable, meaning they can be represented in a basis where they appear as diagonal matrices with non-negative entries. This diagonal form showcases the scaling nature of positive operators:

• The eigenvalues of\( S \) represent the scaling factors.
• These factors are always non-negative.

Positive operators can therefore be viewed as 'magnifying lenses' that stretch vectors without altering their direction.

This behavior plays a role in the polar decomposition of bounded operators, allowing us to distinguish the scaling component from the rotational one.

Bounded Operators

In the context of Hilbert spaces, bounded operators are linear transformations that map vectors without causing them to 'blow up' in length. Formally, an operator \( T \) is bounded if there exists a constant \( C \) such that \( \|T(x)\| \leq C \|x\| \) for all vectors \( x \) in the Hilbert space.

Bounded operators include integral operators, matrix multiplications, and various types of transformations in functional analysis. One key feature is that they map bounded sets to bounded sets.

The boundedness condition is crucial for ensuring that operators behave well in finite-dimensional spaces. In finite dimensions, every linear operator is automatically bounded, but this concept becomes especially important in infinite-dimensional spaces where boundedness must be explicitly checked and guaranteed.

Extreme Points

Extreme points in a convex set are those points that cannot be expressed as a convex combination of other points in that set. They are the 'cornerstones' or 'vertices' of the set. For the unit ball of bounded operators in a Hilbert space, denoted \( B_{\mathcal{B}(H)} \), identifying extreme points involves understanding the structure and properties of the operators.

In this context, unitary operators turn out to be the extreme points. This is because, under the polar decomposition, if an operator can be expressed as\( T = US \), then to remain extreme,\( S \) must act as the identity operator. This means that\( T \) itself should be purely unitary.

The reason is tied to the stability that these unitary operators offer. Their role as extreme points helps in understanding the maximal and minimal boundaries of operator sets in finite-dimensional spaces.