The Battle of Tsushima was a huge Japanese victory over one of the Great Powers of Europe and essentially destroyed the Russian Navy. The victory was instrumental in forcing Russia to negotiate peace with Japan. Did the legacy of this huge victory influence the Japanese to attempt a second victory of this kind over America, feeling that it would have a similar effect? Did they expect a similar response from America as Russia gave? Did they equate the depression in the United States as a similarly weak internal position faced by Russia in 1904 which could be taken advantage of? Is there any connection at all?
The Japanese victory at Tsushima definitely impacted Japanese naval thought for years to come. However, while it did have a role in the decision to attack Pearl Harbor, Pearl Harbor was not the decisive strike they were looking for.
Tsushima was the result of a drawn out and hard fought campaign between the Russian Pacific fleet and the Imperial Japanese Navy. Namely, the loss of the Pacific Fleet at Port Arthur forced Russia to deploy its Baltic Fleet to the Pacific. Meanwhile, the Japanese fleet, led by Togo, was well aware of the Russian movements, and positioned his fleet in home waters, in an area he knew well, near his reinforcements and away from enemy reinforcements, as well as in a tactically favorable position (his ships were positioned to cross the "T" of the incoming Russian ships, meaning that while his battleships and cruisers were able to fire all of their turrets in a broadside, the Russians could only respond with their fore (forward) turrets.). The victory decisively ended the war, as Russia no longer had any naval assets to challenge Japan's control of the Pacific, and no way to convey enough land troops to resist Japanese attack.
Pearl Harbor was hit with a similar goal in mind: however, it was not the decisive strike. The point of the strike was similar to the taking of Port Arthur: namely, to cripple a large amount of the enemy naval assets, and to draw them to mass their forces and attack deep into Japanese waters, being slowly attritioned by submarine screens and aircraft, and eventually allowing Japanese naval assets in home waters, with the support of land-based aircraft, to destroy it in a decisive battle like that at Tsushima, thus ending the war. Ironically, the strike at Pearl Harbor was the nail in the coffin for that sort of strategy (the US "War Plan Orange," which was already on its way out, but became completely impossible due to US battleship losses), and so Japan ended up waiting for the decisive battle that never came. Midway was arguably an attempt to force that battle, but the IJN made some really stupid decisions (AL Operation, MO Operation) that divided their carrier strength to the point that with a little luck (Chikuma's recon plane comes to mind) the US were able to thrash the Japanese carrier strength.
The difference between Tsushima and Pearl Harbor are too huge for one to really affect the other. First off Tsushima was a battle between modern battleships, which by the time of Pearl Harbor were becoming obsolete. Tsushima also wasn't a surprise attack like Pearl Harbor, but rather a drawn out, battle between ships. Pearl Harbor on the other hand was carried out by airplanes launched from Naval Carriers. This isn't meant to downplay the significance of Tsushima, which had a huge effect on naval doctrine of every nation that was paying attention. It also introduced wireless technology as something essential to modern naval combat.
Pearl Harbor was a different beast, it was a surprise attack, done on ships anchored in a harbour. Now there was a battle that did have an effect on Pearl Harbor. This isn't to say Pearl Harbor was an original plan at all, in fact it was heavily based on the battle of Taranto, which was a surprise attack done by the British navy on the Italian fleet located at Taranto. It utterly destroyed the Italian navy and essentially made them a non-factor during the war in the Mediterranean. This is what the Japanese high assumed the attack at Pearl Harbor would do. They hoped to replicate the success the British had at Taranto, but while they did destroy much of the American battleship fleet, they failed to destroy the Carriers, which is a factor the British didn't have to worry about.