Question

Let \(T \in \mathcal{B}(H)\) be a self-adjoint isomorphism of a Hilbert space \(H\) onto \(H .\) Show that if \(T\) is positive (i.e., \((T(x), x) \geq 0\) for all \(x \in X)\), then \([x, y]=(T(x), y)\) defines a new inner product on \(H\) and \(\|x\|=[x, x]^{\frac{1}{2}}\) is an equivalent norm on \(H\).

Short answer

(T(x), y) defines an inner product on H. \|x\|_T = \sqrt{[x, x]} is equivalent to the original norm.

Step-by-step solution

Understand the given properties

Given that 1. \(T \in \mathcal{B}(H)\) is a self-adjoint isomorphism, so \(T = T^*\). 2. \(T\) is positive, meaning \((T(x), x) \geq 0\) for all \(x \in H\).

Define the new inner product

Define a new inner product \([x, y] = (T(x), y)\) on the Hilbert space \(H\).

Verify the new inner product properties

To confirm that \([x, y]\) defines an inner product, we need to check three conditions:1. Linearity in the first argument: \([ax_1 + bx_2, y] = a[x_1, y] + b[x_2, y]\) for any scalars \(a\) and \(b\).2. Symmetry: \([x, y] = \overline{[y, x]}\).3. Positivity: \([x, x] = (T(x), x) \geq 0\).

Check linearity in the first argument

To prove linearity: \([ax_1 + bx_2, y] = (T(ax_1 + bx_2), y) = (aT(x_1) + bT(x_2), y) \) Using the linearity of \(T\) and the original inner product, we get: \([ax_1 + bx_2, y] = a(T(x_1), y) + b(T(x_2), y) = a[x_1, y] + b[x_2, y]\).

Verify symmetry

To show symmetry, use the fact that \(T\) is self-adjoint: \([x, y] = (T(x), y) = \overline{(y, T(x))} = \overline{(T(y), x)} = \overline{[y, x]}\).

Check positivity

By the given positivity condition of \(T\): \([x, x] = (T(x), x) \geq 0\).

Define the equivalent norm

Define \( \|x\|_T = \sqrt{[x, x]} = \sqrt{(T(x), x)}\).

Show norm equivalence

Since \(T\) is an isomorphism, there exist constants \(m, M > 0\) such that \(m\|x\|^2 \leq (T(x), x) \leq M\|x\|^2\). This shows that \( \|x\|_T = \sqrt{(T(x), x)}\) is equivalent to the original norm \( \|x\| \).

Key concepts

Inner Product Spaces

An inner product space is a vector space equipped with an inner product. An inner product is a function that assigns a complex number to every pair of vectors in the space. This function must satisfy specific properties: linearity in the first argument, symmetry, and positivity.

In our context, we define a new inner product \([x, y] = (T(x), y)\) on the Hilbert space \(H\) using a self-adjoint operator \(T\). This means \((T(x), y)\) follows the properties of an inner product:

• **Linearity:** For any scalars \(a\) and \(b\), \[([ax_1 + bx_2, y] = (T(ax_1 + bx_2), y) = a(T(x_1), y) + b(T(x_2), y) = a[x_1, y] + b[x_2, y])\]
• **Symmetry:** Using that \(T\) is self-adjoint, \[([x, y] = (T(x), y) = \overline{(y, T(x))} = \overline{(T(y), x)} = \overline{[y, x]})\]
• **Positivity:** By the given positivity condition of \(T\), \[([x, x] = (T(x), x) \geq 0)\]

Establishing these properties is crucial for defining a valid new inner product on the Hilbert space \(H\).

Positive Operators

Positive operators are fundamental in the study of Hilbert spaces and inner products. An operator \(T\) on a Hilbert space \(H\) is considered positive if it satisfies \((T(x), x) \geq 0\) for all \(x \in H\).

In our example, the positivity of \(T\) helps in defining a new inner product. It ensures that \[([x, x] = (T(x), x) \geq 0)\] which is essential for applying the positivity condition.

Positive operators also imply certain spectral properties which are useful for analyzing and decomposing the operator. This makes them very relevant in quantum mechanics and other areas of functional analysis.

Norm Equivalence

Norm equivalence is about comparing two norms in a vector space and confirming that one can be bounded by a multiple of the other. This implies that the two norms induce the same topology on the space.

For the exercise, we define a new norm by \(\|x\|_T = \sqrt{[x, x]} = \sqrt{(T(x), x)}\). Our goal is to show this new norm is equivalent to the original norm \(\textbackslash\|x\textbackslash\|\).

Because \(T\) is an isomorphism, there exist constants \(m, M > 0\) such that:

• For all \(\|x\|^2 \leq (T(x), x)\right) = [x, x]\leq M\|x\|^2)\).

This means \|x\|_T is equivalent to \|x\|, ensuring both norms respond similarly to the input vectors in terms of scaling and topology.