You mean an infinite-dimensional separable Hilbert space. The answer is no.

Suppose $p(z)$ has distinct roots $\alpha_1, \alpha_2$.  Define a sequence $x_1, x_2, \ldots$ in the unit sphere of $H$ such that 


$x_1,\ldots, x_n$ are linearly independent for all $n$.
$\|x_i - x_{i+1}\| \to 0$ as $i \to \infty$.
the sequence is dense in the unit sphere of $H$.


Define $A$ on the linear span of the sequence so that $A x_i = \alpha_1 x_i$ if $i$ is odd, $\alpha_2 x_i$ otherwise.

On the other hand, if $p$ has only one root, say $p(z) = (z - \alpha)^d$, then 
with the same sequence as above take $A x_i = \alpha x_i + x_{i+1}$ for $i$ not divisible by $d$, $\alpha x_i$ otherwise.

A question on unbounded operators
I would be interested to hear if anyone had recommendations for 2-player games that are particularly suitable for playing while travelling on trains.  (I ask this as someone who mostly enjoys games like Carcassonne, Power Grid, Agricola, Pandemic, etc. around the table at home.)  To be specific, the things that I think are important properties for these games are:


Needing minimal table space.  Most trains have a small table you could use, but not much more.
Being quiet to play, so that other passengers won't be disturbed.  For example, I imagine that games that involve repeatedly rolling dice wouldn't be appreciated.
Packing down small, so that they don't take up much luggage space.
Not requiring batteries or a power socket.  (I appreciate that an iPhone or a Nintendo DS for each player might be a good solution more generally, but it's not what I'm after in this case.)
Not being so delicate to arrange that motion from the train will unduly disrupt the play.


I think it would be best to recommend one game (or class of games) per answer, if that's appropriate, and explain why you think it's particularly suitable.  (Incidentally, I've read the guidance on good questions about game recommendations and hope this question meets the criteria.)

Update: Thanks for so many excellent suggestions - we certainly won't be short of games for our next long train journey :)

Update 2: Unfortunately, I can't really playtest this many great games in any reasonable time period, so I'm going to accept the top-voted answer (Cribbage) and try out the others as soon as I can.  Thanks again...
