20 questions - Can you describe the image of the exponential map B(H)->B(H).

The commenters on the wiki http://scratchpad.wikia.com/wiki/091006qa are very knowledgeable; I'm just expanding on Anonymous's answer.

The exponential map is not surjective. If you look at Halmos's paper http://www.ams.org/mathscinet-getitem?mr=53391 (link requires academic access) you will see a wide variety of invertible maps that are not exponentials. Here is the simplest example. Take 0 < u < v and let D be the annulus u < |z| < v in the complex plane. Our Hilbert space will be the space of analytic functions f on D such that integral |f(z)|^2 is finite. The operator is multiplication by z.

To sketch Halmos' argument, H is complete because the property of being harmonic can be stated as a condition on integrals and thus passes through an L^2 limit. The logarithm of multiplication by z wants to be multiplication by log z, but we can't define log z on D without introducing a branch cut. Of course, D doesn't have to be this exact shape, any open region of the complex plane which has a function without a logarithm would work.