Hellinger-Toeplitz theorem

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Problem 10B.7 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 10B "Spectrum"

Prove that if $T$ is an operator on a Hilbert space $V$ such that $\langle T f, g\rangle = \langle f, T g\rangle$ for all $f, g \in V$ , then $T$ is a bounded operator.

Problem 6.51 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 6.2 "Problems to Solve"

Let $A, B$ be linear operators in the Hilbert space $H$ such that $(A x, y) = (x, B y)$ for all $x, y \in H$ . Prove that $A, B \in \mathcal{L}(H)$ .

Problem 9.29 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 9.2 "Problems to Solve"

Let $A, B$ be linear operators in Hilbert space $H$ such that $(A x, y) = (x, B y), x, y \in H$ . Prove that $A \in \mathcal{L}(H)$ and $B = A^{*}$ .

Problem 7.4 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 7.1 "Banach-Steinhaus Theorem"

(E. Hellinger, O. Toeplitz, 1910). Let $A: H \rightarrow H$ be a linear operator in a Hilbert space $H$ . Prove that if for all $x, y \in H$ the equality $(A x, y) = (x, A y)$ holds, then the operator $A$ is bounded. ¹

Definition 8.3: An operator $A \in \mathcal{B}(H)$ is called self-adjoint if $A = A^{*}$ .

Operators satisfying such a relation are called self-adjoint (see Definition 8.3 below). ↩

Problem 60 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 6 "Boundedness and Invertibility"

(a)(b)

(a)

Is a symmetric linear transformation on an inner-product space $\mathbf{H}$ necessarily bounded?

(b)

What if $\mathbf{H}$ is a Hilbert space?

Remark.

A linear transformation $A$ (not necessarily bounded) on an inner-product space $\mathbf{H}$ (not necessarily complete) is called symmetric if $\left\langle Af, g \right\rangle = \left\langle f, Ag \right\rangle$ for all $f$ and $g$ in $\mathbf{H}$ . (This neutral term is used rather than "Hermitian" or "self-adjoint," because in the customary approach to Hermitian operators, $A = A^{*}$ , boundedness is an assumption necessary for the very formulation of the definition.)

Problem 78 from Le A. "Functional Analysis Problems with Solutions" [Le13]More problems in Chapter 5 "Fundamental Theorems"

Let $H$ be a Hilbert space and $T: H \rightarrow H$ a linear operator which is symmetric, i.e.,

\langle T x, y\rangle = \langle x, T y\rangle , \forall x, y \in H

Prove that $T$ is continuous.

Problem 11.3.8 from Petrovic J.S. "Measure and Integral: Theory and Practice" [Pet25]More problems in Section 11.3 "The Uniform Boundedness Principle"

Let $\mathcal{H}$ be Hilbert space and let $A, B: \mathcal{H} \rightarrow \mathcal{H}$ be linear operators satisfying $\left\langle B x, y \right\rangle = \left\langle x, A y \right\rangle$ for all $x, y \in \mathcal{H}$ . Prove that $A$ and $B$ are both bounded.

Problem 11.4.44 from Petrovic J.S. "Measure and Integral: Theory and Practice" [Pet25]More problems in Section 11.4 "The Open Mapping and Closed Graph Theorems"

Let $\mathcal{H}$ be Hilbert space, let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a linear operator that is symmetric, i.e., $\left\langle T x, y \right\rangle = \left\langle x, T y \right\rangle$ , for all $x, y \in \mathcal{H}$ . Prove that $T$ is bounded. Give an example to show that the assertion is not true if $D$ is a dense subspace of $\mathcal{H}$ and $T: D \rightarrow \mathcal{H}$ is symmetric.

Problem 11.4.45 from Petrovic J.S. "Measure and Integral: Theory and Practice" [Pet25]More problems in Section 11.4 "The Open Mapping and Closed Graph Theorems"

Let $\mathcal{H}$ be Hilbert space, let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a linear operator such that $\left\langle T x, x \right\rangle \geq 0$ , for all $x \in \mathcal{H}$ . Prove that $T$ is bounded.

Problem 20.30 from Royden H.L., Fitzpatrick P.M. "Real Analysis" [RoFi23]More problems in Section 20.4 "Symmetric Operators"

Let $H$ be a Hilbert space and the linear operator $T: H \rightarrow H$ have the property that $\left\langle T(u), v \right\rangle = \left\langle u, T(v) \right\rangle$ for all $u, v \in H$ . Show that $T$ belongs to $\mathcal{L}(H)$ , that is, $T$ is bounded.

References

[Axl19] Axler S. Measure, Integration & Real Analysis. 3rd ed., Springer, 2019.

[BCK+24] Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. Functional Analysis and Operator Theory. Springer, 2024.

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.

[Hal82] Halmos P.R. A Hilbert Space Problem Book. 2nd ed., Springer-Verlag, 1982.

[Le13] Le A. Functional Analysis Problems with Solutions. 2013.

[Pet25] Petrovic J.S. Measure and Integral: Theory and Practice. CRC Press, 2025.

[RoFi23] Royden H.L., Fitzpatrick P.M. Real Analysis. 5th ed., Pearson, 2023.

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